#149729

order	512
nilpotency class	3
isoclinism	isoclinic class of #1510
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

237 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 86 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)(9,10,14,21,16,23,38,50,17,20,34,19,39,48,62,49,15,22,32,47,31,46,63,69,13,36,12,30,33,60,35,61)(11,24,27,51,29,52,59,70,26,18,40,42,57,44,64,68,28,45,58,41,54,66,72,67,37,43,25,53,55,65,56,71);(2,5)(3,7)(9,11)(10,18)(12,25)(13,37)(14,27)(15,28)(16,29)(17,26)(19,41)(20,45)(21,42)(22,43)(23,44)(24,36)(30,51)(31,54)(32,58)(33,55)(34,40)(35,56)(38,59)(39,57)(46,65)(47,53)(48,66)(49,67)(50,68)(52,60)(61,70)(62,64)(63,72)(69,71);(9,12,31,62,17,14,33,63,15,34,16,35,13,32,39,38)(10,19,46,61,20,47,60,50,22,30,23,49,36,21,48,69)(11,25,54,64,26,27,55,72,28,40,29,56,37,58,57,59)(18,41,65,70,45,53,52,68,43,51,44,67,24,42,66,71)

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,c^a=ci,d^a=df,d^b=di,e^b=efi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56925);