#498800

order	512
nilpotency class	4
isoclinism	isoclinic class of #498800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#469
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1999
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1519 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)(5,21)(6,26)(7,11)(8,15)(10,24)(12,23)(14,40)(16,39)(17,45)(18,32)(19,48)(20,51)(22,37)(25,53)(27,35)(28,56)(29,44)(30,54)(31,42)(33,50)(34,47)(36,55)(38,41)(43,61)(46,62)(49,57)(52,60)(58,63)(59,64);(1,3)(2,8)(4,14)(5,22)(6,27)(7,16)(9,31)(10,35)(11,29)(12,37)(13,21)(15,24)(17,23)(18,41)(19,49)(20,52)(25,54)(26,30)(28,32)(33,60)(34,57)(36,45)(38,62)(39,53)(40,48)(42,50)(43,47)(44,55)(46,63)(51,58)(56,64)(59,61);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,28)(10,33)(11,32)(12,34)(13,38)(15,42)(16,41)(17,43)(21,48)(22,49)(23,47)(24,50)(25,46)(26,51)(27,52)(29,31)(30,57)(35,61)(36,59)(37,58)(39,40)(44,56)(45,60)(53,62)(54,63)(55,64);(4,18)(9,32)(14,41)(19,46)(20,47)(28,31)(33,59)(34,51)(38,40)(42,56)(43,52)(48,62)(49,63)(50,64)(57,58)(60,61)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249827, #72249830, #72249832, …
maximal normal subgroup of	#72249827, #72249830, #72249832, …
maximal quotient of	#72257110, #72257126, #72322962

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405996);