#6469082

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	#34963
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1701 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)(3,17)(4,23)(5,12)(6,13)(7,15)(8,14)(9,16)(10,21)(11,26)(18,52)(19,50)(20,51)(22,47)(24,35)(25,45)(27,48)(28,41)(29,40)(30,44)(31,46)(32,39)(33,38)(34,43)(36,58)(37,59)(42,57)(49,56)(53,63)(54,62)(55,61)(60,64);(1,3)(2,10)(4,24)(5,19)(6,20)(7,33)(8,21)(9,22)(11,35)(12,42)(13,37)(14,47)(15,17)(16,38)(18,27)(23,49)(25,59)(26,55)(28,56)(29,58)(30,54)(31,53)(32,60)(34,57)(36,44)(39,63)(40,50)(41,52)(43,62)(45,64)(46,51)(48,61);(1,4,8,26,9,27,7,28)(2,11,15,23,16,41,14,48)(3,18,21,56,22,24,33,55)(5,25,29,32,30,31,34,6)(10,35,17,49,38,52,47,61)(12,39,43,45,44,13,40,46)(19,53,58,20,54,59,57,60)(36,37,50,51,42,63,62,64);(1,5)(2,12)(3,19)(4,25)(6,28)(7,34)(8,29)(9,30)(10,36)(11,39)(13,41)(14,40)(15,43)(16,44)(17,50)(18,53)(20,56)(21,58)(22,54)(23,45)(24,59)(26,32)(27,31)(33,57)(35,37)(38,42)(46,48)(47,62)(49,51)(52,63)(55,60)(61,64);(1,6,7,31,9,32,8,25)(2,13,15,46,16,39,14,45)(3,20,33,53,22,60,21,59)(4,5,28,34,27,30,26,29)(10,37,17,51,38,63,47,64)(11,40,23,12,41,43,48,44)(18,54,55,58,24,19,56,57)(35,50,49,42,52,62,61,36)

Polycyclic presentation:

2^9|c^2=g,e^2=gh,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dhi,f^b=fi,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285642
maximal normal subgroup of	#72285642

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376278);