#6894266

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1500 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)(17,18,23,27,22,29,21,28)(19,24,37,40,33,42,32,41)(20,26,34,25,36,45,35,43)(30,44,46,38,31,39,47,48);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,30)(21,32)(22,33)(23,37)(25,38)(26,44)(27,40)(28,41)(29,42)(31,36)(34,46)(35,47)(39,45)(43,48);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,30)(21,34)(22,36)(23,35)(24,38)(26,27)(28,45)(29,43)(31,33)(32,46)(37,47)(39,41)(40,44)(42,48);(2,6)(5,7)(11,16)(13,15)(17,20)(18,26)(19,31)(21,35)(22,36)(23,34)(24,39)(25,27)(28,43)(29,45)(30,33)(32,46)(37,47)(38,41)(40,48)(42,44);(10,14)(13,15)(17,21,22,23)(18,27,29,28)(19,32,33,37)(20,34,36,35)(24,40,42,41)(25,26,43,45)(30,46,31,47)(38,44,48,39)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6801462);