#7142605

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1195 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,28,32,29,33,38,40)(27,34,37,39,36,30,35,31);(1,3,5,8)(2,6,4,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,30)(28,37)(29,36)(31,32)(33,34)(35,38)(39,40);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27)(26,31)(28,35)(29,36)(30,32)(33,39)(34,40)(37,38);(1,2,5,4)(3,7,8,6)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,27)(26,30)(28,37)(29,36)(31,32)(33,34)(35,38)(39,40);(1,4,5,2)(3,7,8,6)(10,16)(14,17)(15,24)(22,23)(25,28,29,38)(26,32,33,40)(27,35,36,37)(30,39,34,31)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fi,c^2=h,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,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, 7049801);