#6605402

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1510 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)(19,28)(20,32)(21,27)(22,26)(23,36)(24,30)(25,33)(29,44)(31,40)(34,42)(35,39)(37,41)(38,46)(43,47)(45,48);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,29)(21,35)(22,30)(23,31)(25,38)(26,28)(27,40)(32,48)(33,47)(34,45)(36,39)(37,46)(41,43)(42,44);(1,3,5,4)(2,7,8,6)(10,13)(11,15)(17,20)(18,25)(19,29)(21,33)(22,34)(23,37)(24,38)(26,41)(27,42)(28,43)(30,45)(31,46)(32,36)(35,47)(39,48)(40,44);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(18,26)(19,30)(25,41)(27,36)(29,45)(31,35)(32,42)(46,47);(1,4,5,3)(2,6,8,7)(10,13)(11,15)(17,21,22,23)(18,27,26,36)(19,31,30,35)(20,33,34,37)(24,39,28,40)(25,42,41,32)(29,46,45,47)(38,48,43,44)

Polycyclic presentation:

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

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, 6512598);