#417686

order	512
nilpotency class	3
isoclinism	isoclinic class of #417686
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15257
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2344 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:

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

Polycyclic presentation:

2^9|e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^c=hi,e^d=ei,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, 324882);