#7188971

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1328 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=i,d^2=fg,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7096167);