#7622438

order	512
nilpotency class	3
isoclinism	isoclinic class of #3585
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂³×D₄
commutator	C₄	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂	C₂³×C₂²⋊C₈
prime radical	C₂×#585	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529634);