#500683

order	512
nilpotency class	4
isoclinism	isoclinic class of #499952
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3205
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#24208
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1161 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)(3,15)(4,21)(5,13)(6,28)(7,30)(8,14)(9,18)(10,22)(11,26)(12,29)(16,47)(17,53)(19,42)(20,46)(23,43)(24,58)(25,38)(27,44)(31,35)(32,39)(33,61)(34,54)(36,55)(37,56)(40,60)(41,59)(45,50)(48,62)(49,57)(51,64)(52,63);(1,3,6,18,8,20,29,55)(2,9,12,15,14,36,28,46)(4,16,23,49,25,52,24,51)(5,17,26,19,27,54,7,31)(10,37,39,62,40,33,41,45)(11,34,13,42,30,53,44,35)(21,57,58,47,38,64,43,63)(22,48,59,56,60,50,32,61);(1,4)(2,10)(3,16)(5,22)(6,23)(7,32)(8,25)(9,33)(11,38)(12,39)(13,43)(14,40)(15,45)(17,48)(18,49)(19,56)(20,52)(21,30)(24,29)(26,59)(27,60)(28,41)(31,61)(34,57)(35,63)(36,37)(42,47)(44,58)(46,62)(50,54)(51,55)(53,64);(1,5,6,26,8,27,29,7)(2,11,12,13,14,30,28,44)(3,17,18,19,20,54,55,31)(4,22,23,59,25,60,24,32)(9,34,15,42,36,53,46,35)(10,38,39,43,40,21,41,58)(16,48,49,56,52,50,51,61)(33,57,45,47,37,64,62,63)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307828, #72307884, #72307945
maximal normal subgroup of	#72307828, #72307884, #72307945

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407879);