#149728

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	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×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

241 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 90 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)(9,10,14,21,16,23,37,51,17,20,33,47,38,49,60,68,15,22,31,50,30,45,61,69,13,35,12,19,32,46,34,48)(11,24,27,40,29,53,59,65,26,18,39,41,57,43,62,66,28,44,58,64,54,67,71,72,36,42,25,52,55,63,56,70);(2,5)(3,7)(9,11)(10,18)(12,25)(13,36)(14,27)(15,28)(16,29)(17,26)(19,40)(20,44)(21,41)(22,42)(23,43)(24,35)(30,54)(31,58)(32,55)(33,39)(34,56)(37,59)(38,57)(45,63)(46,53)(47,64)(48,65)(49,67)(50,52)(51,66)(60,62)(61,71)(68,72)(69,70);(9,12,30,60,17,14,32,61,15,33,16,34,13,31,38,37)(10,19,45,68,20,21,46,69,22,47,23,48,35,50,49,51)(11,25,54,62,26,27,55,71,28,39,29,56,36,58,57,59)(18,40,63,72,44,41,53,70,42,64,43,65,24,52,67,66)

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,d^b=di,e^b=efi

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, 56924);