#6602897

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1379 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6510093);