#493677

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1172 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:

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

Polycyclic presentation:

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

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, 400873);