#72343220

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×#120
commutator	C₂⁴	C₂²×C₄²
Frattini, cosocle	C₂²×C₄²	C₂⁴
socle	C₂²	C₄×#386
prime radical	C₂×#525	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

3129 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,6,4,5,7,8)(9,10,13,17,14,18,19,26)(11,15,21,28,22,29,32,37)(12,23,24,36,25,16,30,31)(20,33,34,40,35,27,38,39);(2,5)(6,8)(9,11,19,32,14,22,13,21)(10,15,26,37,18,29,17,28)(12,20,30,38,25,35,24,34)(16,27,36,40,23,33,31,39);(9,12)(10,16)(11,20)(13,24)(14,25)(15,27)(17,31)(18,23)(19,30)(21,34)(22,35)(26,36)(28,39)(29,33)(32,38)(37,40);(16,23)(27,33)(31,36)(39,40)

Polycyclic presentation:

2^10|a^2=e,b^2=ehj,e^2=i,h^2=j,b^a=bj,c^a=cij,d^a=df,d^c=dg,f^c=fij,g^a=gij

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343222, #72343263
maximal normal subgroup of	#72343222, #72343263

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