#6598899

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1411 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:

(1,2,5,7)(3,8,4,6)(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);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(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);(1,4,5,3)(2,6,7,8)(10,13)(11,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,34)(27,40)(28,41)(29,43)(31,45)(32,46)(33,47)(39,44)(42,48);(9,11)(10,14)(12,15)(13,16)(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);(10,13)(11,15)(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=fi,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, 6506095);