#7622434

order	512
nilpotency class	3
isoclinism	isoclinic class of #2883
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂²×C₂²⋊C₄
commutator	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂	C₂³×C₂²⋊C₈
prime radical	C₂×#585	C₄
LCS	G ▷ C₂² ▷ 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:

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

Polycyclic presentation:

2^9|a^2=g,g^2=h,h^2=i,b^a=bf,f^a=fi,e^b=ei,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, 7529630);