#492853

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400049);