#149667

order	512
nilpotency class	3
isoclinism	isoclinic class of #1510
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

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

normal subgroups only

This group has 254 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) = 128

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

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

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,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, 56863);