#6597419

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1701 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504615);