#72321353

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23960
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#498869	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20435 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2240 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:

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

Polycyclic presentation:

2^10|a^2=e,c^2=f,d^2=f,h^2=i,i^2=j,b^a=bh,c^b=cf,d^b=dg,d^c=dej,e^b=ej,g^c=gj,h^a=hi,h^b=hij,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321390, #72322673
maximal normal subgroup of	#72321390, #72322673

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