#7194000

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1479 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=fg,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=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, 7101196);