#7193851

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1179 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7101047);