#72250987

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#486
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂²	C₂×#2041
prime radical	#6630018	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2703 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=gi,d^2=eg,e^2=i,h^2=j,i^2=j,c^a=cghj,c^b=cf,d^a=dh,d^c=de,e^a=ej,e^c=ei,e^d=ej,g^c=gj,g^d=gj,i^c=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251004, #72252755, #72256849
maximal normal subgroup of	#72251004, #72252755, #72256849

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.7);;
gap> SetPower(coll, 3, F.7*F.9);;
gap> SetPower(coll, 4, F.5*F.7);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);