#7181892

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	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1420 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)(12,20)(15,21)(19,24)(22,23)(25,26,31,36)(27,32,30,35)(28,38,29,34)(33,39,40,37);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,31,30)(26,32,36,35)(28,37,29,39)(33,34,40,38);(1,2,5,4)(3,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,28,31,29)(26,33,36,40)(27,37,30,39)(32,34,35,38);(1,4,5,2)(3,6,8,7)(11,18)(14,17)(19,24)(22,23)(25,29,31,28)(26,34,36,38)(27,37,30,39)(32,33,35,40);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(25,27,31,30)(26,35,36,32)(28,37,29,39)(33,38,40,34)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=fi,d^2=fi,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, 7089088);