#94604

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₃₂
Frattini, cosocle	C₄×C₃₂	C₂²
socle	C₂²	C₂²⋊C₃₂
prime radical	C₂×D₄	C₃₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

78 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2)(3,5)(4,7)(6,8)(9,10,11,17,13,18,22,35,14,19,23,36,28,38,44,57,15,20,24,37,29,39,45,58,30,40,46,59,50,60,64,70,12,16,21,31,25,32,41,51,26,33,42,52,47,54,61,66,27,34,43,53,48,55,62,67,49,56,63,68,65,69,71,72);(2,5)(3,8)(6,7)(10,16)(11,21)(18,32)(19,33)(20,34)(22,41)(23,42)(24,43)(38,54)(39,55)(40,56)(44,61)(45,62)(46,63)(60,69)(64,71)

Polycyclic presentation:

2^9|a^2=d,c^2=ei,d^2=f,f^2=g,g^2=h,h^2=i,b^a=bc,c^a=ce,c^b=cei,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, 1800);