#499396

order	512
nilpotency class	4
isoclinism	isoclinic class of #498800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#469
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2076
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

667 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320665, #72320677, #72320688, …
maximal normal subgroup of	#72320665, #72320677, #72320688, …
maximal quotient of	#72322809, #72322832, #72322869

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406592);