#6986686

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

1332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,d^2=fhi,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, 6893882);