#7194067

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1348 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)(26,30)(27,31)(32,35)(33,38)(34,37)(39,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,21)(10,16,12,18)(13,20,14,19)(17,24,22,23)(25,26,29,30)(27,31,34,37)(28,35,36,32)(33,39,40,38);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,16,21,18)(13,17,14,22)(19,24,20,23)(25,27,26,31,29,34,30,37)(28,33,32,38,36,40,35,39);(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)(26,32)(27,33)(29,36)(30,35)(31,38)(34,40)(37,39);(9,14,15,13)(10,17,12,22)(11,20,21,19)(16,23,18,24)(27,34)(28,36)(31,37)(32,35)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=fgi,d^2=fi,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=dh,e^c=ei,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, 7101263);