#6894198

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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1507 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)(17,18,21,27,22,28,23,29)(19,30,38,42,34,24,33,45)(20,35,39,25,37,26,36,43)(31,44,47,46,32,41,48,40);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,31)(21,38)(22,34)(23,33)(25,40)(26,44)(27,45)(28,30)(29,42)(32,37)(35,41)(36,48)(39,47)(43,46);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,22,37)(18,25,28,43)(19,31,34,32)(21,36,23,39)(24,40,30,46)(26,27,35,29)(33,47,38,48)(41,42,44,45);(2,6)(5,7)(11,16)(13,15)(17,20)(18,26)(19,32)(21,39)(22,37)(23,36)(24,41)(25,29)(27,43)(28,35)(30,44)(31,34)(33,47)(38,48)(40,45)(42,46);(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,33,34,38)(20,36,37,39)(24,42,30,45)(25,35,43,26)(31,47,32,48)(40,44,46,41)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=i,e^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, 6801394);