#7107609

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#31036	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1371 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=hi,d^2=i,e^2=f,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=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7014805);