#7169743

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1194 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)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,31,29)(27,30,34,32)(28,35,38,36)(33,40,39,37);(1,3,5,8)(2,6,4,7)(11,17)(12,20)(14,21)(15,19)(25,27)(26,32)(28,36)(29,30)(31,34)(33,37)(35,38)(39,40);(1,2,5,4)(3,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,28,31,38)(26,33,29,39)(27,35,34,36)(30,37,32,40);(1,4,5,2)(3,6,8,7)(11,17)(14,21)(18,23)(22,24)(25,29,31,26)(27,30,34,32)(28,33,38,39)(35,40,36,37);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,30,31,32)(26,34,29,27)(28,37,38,40)(33,36,39,35)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=f,d^2=f,e^2=h,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, 7076939);