#7186418

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1250 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)(12,20)(15,21)(19,24)(22,23)(25,26)(27,32)(28,37)(29,33)(30,40)(31,34)(35,39)(36,38);(2,5)(3,7)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,31,30)(26,32,34,40)(28,33,38,35)(29,36,39,37);(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,28,30,33,31,38,27,35)(26,29,32,37,34,39,40,36);(1,3,4,7)(2,6,5,8)(11,18)(14,17)(19,24)(22,23)(25,29,31,39)(26,33,34,35)(27,36,30,37)(28,32,38,40);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(26,34)(29,39)(32,40)(36,37)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=fi,e^2=hi,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,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7093614);