#495336

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,d^c=di

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, 402532);