#7622563

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#117
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂	C₂²×#432
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1465 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529759);