#409281

order	512
nilpotency class	3
isoclinism	isoclinic class of #394328
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2617
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15014
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

963 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:

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

Polycyclic presentation:

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

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, 316477);