#7035307

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,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, 6942503);