#149666

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

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	#525	C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 260 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) = 128

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

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

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,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, 56862);