#7180382

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1278 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)(9,10,14,17)(11,18,12,15)(13,21,22,16)(19,23,20,24)(25,26)(27,30)(28,31)(29,33)(32,36)(34,38)(35,39)(37,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,14)(13,20)(16,24)(17,18)(19,22)(21,23)(25,27,29,34)(26,30,33,38)(28,35,36,40)(31,39,32,37);(1,4,5,2)(3,7,8,6)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(26,31)(27,34)(28,36)(30,37)(32,33)(38,39);(3,8)(6,7)(11,12)(15,18)(19,20)(23,24)(25,27)(26,30)(28,35)(29,34)(31,39)(32,37)(33,38)(36,40);(9,13,14,22)(10,16,17,21)(11,19,12,20)(15,23,18,24)(27,34)(30,38)(35,40)(37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,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, 7087578);