#499302

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×#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

751 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)(3,16,22,49)(4,23,25,43)(5,13)(6,31)(7,34)(8,15)(9,18,39,55)(10,27,41,60)(11,29)(12,30)(14,33)(17,50)(19,38,20,57)(21,45,56,37)(24,48)(26,47,32,44)(28,42,35,46)(36,54)(40,53)(51,63)(52,64)(58,62)(59,61);(1,3,5,18,7,21,29,19,8,22,30,55,33,56,6,20)(2,9,11,16,14,38,13,37,15,39,31,49,34,57,12,45)(4,24,26,51,28,58,27,59,25,53,32,17,35,52,60,54)(10,40,42,36,44,62,23,50,41,48,46,61,47,64,43,63);(1,4)(2,10)(3,17)(5,27)(6,32)(7,35)(8,25)(9,36)(11,23)(12,46)(13,42)(14,47)(15,41)(16,48)(18,24)(19,52)(20,58)(21,59)(22,51)(26,29)(28,33)(30,60)(31,43)(34,44)(37,62)(38,63)(39,61)(40,49)(45,64)(50,57)(53,55)(54,56);(4,25)(10,41)(17,51)(23,43)(24,53)(26,32)(27,60)(28,35)(36,61)(40,48)(42,46)(44,47)(50,63)(52,58)(54,59)(62,64)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,e^2=h,f^2=hi,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=fh,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, 406498);