#501195

order	512
nilpotency class	4
isoclinism	isoclinic class of #501188
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24404
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1066 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)(4,10)(5,24)(6,26,28,12)(7,13)(8,14)(9,18,34,51)(11,25,41,29)(16,44)(17,33)(19,45)(20,52)(21,39)(22,55,27,38)(23,40)(30,59)(31,43)(32,47,61,50)(35,49,42,58)(36,53,62,57)(37,54,63,56)(46,60)(48,64);(1,3,8,19)(2,9,14,34)(4,20,23,46)(5,17,7,30)(6,18,28,51)(10,36,40,62)(11,33,41,59)(12,15,26,45)(13,42,24,35)(16,31,48,21)(22,53,27,57)(25,49,29,58)(32,43,61,39)(37,44,63,64)(38,52,55,60)(47,56,50,54);(1,4,8,23)(2,10,14,40)(3,16,19,48)(5,21,7,31)(6,27,28,22)(9,32,34,61)(11,37,41,63)(12,38,26,55)(13,43,24,39)(15,44,45,64)(17,46,30,20)(18,50,51,47)(25,56,29,54)(33,60,59,52)(35,62,42,36)(49,53,58,57);(1,5,8,7)(2,11,14,41)(3,17,19,30)(4,21,23,31)(6,25,28,29)(9,33,34,59)(10,37,40,63)(12,24,26,13)(15,35,45,42)(16,46,48,20)(18,49,51,58)(22,54,27,56)(32,60,61,52)(36,44,62,64)(38,39,55,43)(47,57,50,53)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303725, #72303728, #72303731, …
maximal normal subgroup of	#72303725, #72303728, #72303731, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408391);