#6894215

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1541 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6801411);