#6993090

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1452 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)(17,18,23,28)(19,24,30,39)(20,32,34,44)(21,27,22,26)(25,37,42,35)(29,45,33,48)(31,43,36,40)(38,46,41,47);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,15)(13,16)(17,19,23,30)(18,24,28,39)(20,33,34,29)(21,36,22,31)(25,41,42,38)(26,43,27,40)(32,48,44,45)(35,46,37,47);(3,8)(5,7)(9,11,14,10)(12,15,13,16)(17,20,23,34)(18,25,28,42)(19,29,30,33)(21,35,22,37)(24,38,39,41)(26,32,27,44)(31,46,36,47)(40,48,43,45);(2,6)(5,7)(9,12,14,13)(10,15,11,16)(19,30)(20,34)(24,39)(25,42)(31,36)(32,44)(35,37)(40,43);(9,13,14,12)(10,15,11,16)(17,21,23,22)(18,26,28,27)(19,31,30,36)(20,35,34,37)(24,40,39,43)(25,32,42,44)(29,46,33,47)(38,48,41,45)

Polycyclic presentation:

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

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, 6900286);