#511768

order	512
nilpotency class	4
isoclinism	isoclinic class of #511767
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32878
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3059 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418964);