#499308

order	512
nilpotency class	4
isoclinism	isoclinic class of #498802
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×#2065
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

747 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,e^2=h,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,h^a=hi,c^b=cg,f^b=fi,d^c=di,e^c=eh,f^c=fhi,g^c=gh,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, 406504);