#149668

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	#525	C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

257 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=g,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,d^a=df,c^b=ci,d^b=di,e^b=efi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56864);