#499366

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×#2073
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

677 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)(3,15,19,9)(4,20,22,44)(5,12)(6,13)(7,14)(8,11)(10,23,36,31)(16,47)(17,43,30,33)(18,46,52,34)(21,54)(24,41,57,64)(25,39)(26,58,59,40)(27,60,61,37)(28,42)(29,38)(32,49)(35,56)(45,50)(48,62)(51,63)(53,55);(1,3)(2,9)(4,21)(5,17)(6,26)(7,30)(8,19)(10,35)(11,15)(12,33)(13,40)(14,43)(16,24)(18,28)(20,53)(22,55)(23,56)(25,52)(27,49)(29,59)(31,51)(32,37)(34,42)(36,63)(38,58)(39,46)(41,62)(44,54)(45,60)(47,64)(48,57)(50,61);(1,4)(2,10,11,36)(3,16)(5,23)(6,27)(7,31)(8,22)(9,32,15,45)(12,20,14,44)(13,41,38,64)(17,49)(18,51)(19,48)(21,26)(24,28)(25,57)(29,61)(30,50)(33,62,43,47)(34,54,46,53)(35,58,63,40)(37,39,60,42)(52,56)(55,59);(4,22)(10,36)(16,48)(20,44)(21,55)(23,31)(24,57)(27,61)(32,45)(35,63)(37,60)(41,64)(47,62)(49,50)(51,56)(53,54)

Polycyclic presentation:

2^9|a^2=fh,c^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,g^b=gh,h^b=hi,d^c=di,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, 406562);