#6564495

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535687
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	D₄×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,6,9,12)(4,7,11,14)(10,13,15,16)(17,18,23,28)(19,29,33,39)(20,34,36,41)(21,26,22,27)(24,32,40,31)(25,37,42,35)(30,44,46,38)(43,47,48,45);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,22,32,23,33,21,31)(18,24,26,39,28,40,27,29)(20,30,37,47,36,46,35,45)(25,38,41,48,42,44,34,43);(3,9)(6,12)(10,15)(13,16)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,41)(27,34)(28,42)(29,43)(31,45)(32,47)(33,46)(39,48)(40,44);(1,4)(2,7)(3,10)(5,11)(6,13)(8,14)(9,15)(12,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,39,40,29)(25,41,42,34)(30,45,46,47)(38,48,44,43);(4,11)(7,14)(10,15)(13,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,39,40,29)(25,41,42,34)(30,45,46,47)(38,48,44,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,g^c=gi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6471691);