#499520

order	512
nilpotency class	4
isoclinism	isoclinic class of #499029
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#482
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2092
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

721 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406716);