#6595356

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1293 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,32)(21,26)(22,27)(23,38)(24,30)(25,35)(29,45)(31,41)(33,48)(34,44)(36,40)(37,43)(39,46)(42,47);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,33)(21,36)(22,30)(23,31)(25,42)(26,41)(27,28)(29,34)(32,45)(35,46)(37,47)(38,40)(39,43)(44,48);(11,15)(13,16)(17,20,23,35,22,34,21,37)(18,25,26,32,27,43,38,44)(19,29,31,47,30,33,36,46)(24,39,41,48,28,42,40,45);(1,3,5,4)(2,7,8,6)(9,11)(10,13)(12,15)(14,16)(19,30)(20,34)(24,28)(25,43)(31,36)(32,44)(35,37)(40,41);(1,4,5,3)(2,6,8,7)(11,15)(13,16)(17,21,22,23)(18,26,27,38)(19,31,30,36)(20,35,34,37)(24,40,28,41)(25,44,43,32)(29,46,33,47)(39,48,42,45)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6502552);