#6484590

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 6391786);