#6894237

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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1522 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)(17,18,22,26,21,25,34,40)(19,27,30,43,29,23,36,38)(20,31,33,39,32,41,35,24)(28,44,46,37,45,48,47,42);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,29)(18,23,25,27)(20,28,32,45)(22,30,34,36)(24,37,39,42)(26,38,40,43)(31,48,41,44)(33,46,35,47);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,21,32)(18,24,25,39)(19,28,29,45)(22,35,34,33)(23,37,27,42)(26,41,40,31)(30,47,36,46)(38,44,43,48);(2,6)(5,7)(11,16)(13,15)(18,25)(19,29)(20,32)(26,40)(30,36)(33,35)(37,42)(44,48);(10,14)(13,15)(18,25)(19,29)(24,39)(26,40)(28,45)(30,36)(31,41)(46,47)

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^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

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, 6801433);