#6963351

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1389 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)(3,6)(4,5)(7,8)(11,17)(14,18)(19,20)(22,23)(25,26,30,34,29,33,28,38)(27,35,37,40,36,31,39,32);(1,3,7,4)(2,5,8,6)(9,10)(11,18)(12,15)(13,16)(14,17)(19,23)(20,22)(21,24)(25,27)(26,31)(28,39)(29,36)(30,37)(32,34)(33,35)(38,40);(1,4,7,3)(2,5,8,6)(9,11,13,17)(10,14,16,18)(12,20,21,19)(15,23,24,22)(25,27)(26,32)(28,37)(29,36)(30,39)(31,34)(33,40)(35,38);(1,5,7,6)(2,3,8,4)(9,12,13,21)(10,15,16,24)(11,19,17,20)(14,22,18,23)(26,33)(27,36)(34,38)(37,39);(1,6,7,5)(2,3,8,4)(10,16)(14,18)(15,24)(22,23)(25,28,29,30)(26,34,33,38)(27,37,36,39)(31,40,35,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=fhi,d^2=fhi,e^2=f,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, 6870547);