#149730

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂×C₄

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	#532	C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 239 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=fg,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, 56926);