#6913770

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907977
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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6820966);