#135726

order	512
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

(2,4)(5,6,7,9,8,10,11,12)(13,14)(15,23)(16,28)(17,20)(18,31)(19,25)(21,24)(22,29)(26,35)(27,42)(30,43)(32,41)(33,44)(34,39)(36,38)(37,40);(13,15,24,38,18,27,30,41,17,26,39,44,29,40,16,25)(14,19,28,37,22,33,34,35,20,32,43,42,31,36,21,23);(1,2,3,4)(14,20)(19,32)(21,34)(22,31)(23,35)(28,43)(33,36)(37,42)

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=ei,d^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42922);