#7105607

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#42685	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1443 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=fhi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7012803);