#128545

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

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

396 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgi,c^b=ci,d^b=di

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, 35741);