#501335

order	512
nilpotency class	4
isoclinism	isoclinic class of #501334
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24506
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

656 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408531);