#72250915

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23708
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#13323	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,9,16,10,18,20,30,12,7,19,14,11,17,13,27)(8,22,24,33,23,15,31,34,26,35,32,36,25,21,28,29);(5,7,20,30,11,18,9,16,12,6,13,27,10,17,19,14)(8,22,31,29,25,21,24,36,26,35,28,34,23,15,32,33);(2,4)(6,14,7,16)(8,23)(9,20)(10,11)(13,19)(15,33,21,36)(17,27,18,30)(22,34,35,29)(24,32)(25,26);(1,2,3,4)(5,8)(6,15,7,21)(9,24)(10,23)(11,25)(12,26)(13,28)(14,29,16,34)(17,22,18,35)(19,32)(20,31)(27,33,30,36)

Polycyclic presentation:

2^10|a^2=e,b^2=ehj,c^2=hi,d^2=fij,e^2=i,f^2=j,g^2=j,h^2=j,i^2=j,b^a=bij,c^a=cegj,c^b=cegj,d^a=df,d^b=dh,d^c=dfgh,e^b=ej,e^c=ei,f^a=fj,f^b=fj,f^c=fj,g^c=gj,g^d=gj,h^a=hj,h^b=hj,h^c=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250937, #72257311, #72257453
maximal normal subgroup of	#72250937, #72257311, #72257453

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.5*F.8*F.10);;
gap> SetPower(coll, 3, F.8*F.9);;
gap> SetPower(coll, 4, F.6*F.9*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);