#72265095

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#2471
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
socle	C₂²	#15669
prime radical	C₂³×#41	C₂³
LCS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

3933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=e,b^2=g,c^2=i,d^2=h,e^2=j,b^a=bej,c^b=cei,d^a=dg,d^b=dh,d^c=df,e^b=ej,f^a=fj,g^c=gj,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265167, #72265627, #72314310
maximal normal subgroup of	#72265167, #72265627, #72314310

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