#6990470

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1613 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=i,d^2=fh,e^2=fhi,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, 6897666);