#7194217

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

1391 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=fgi,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=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, 7101413);