#6728852

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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

1192 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)(3,6)(4,7)(5,8)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26)(27,32)(28,33)(29,37)(30,34)(31,35)(36,40)(38,39);(1,3,5,4)(2,6,8,7)(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24)(25,27,31,37)(26,29,35,32)(28,33,38,39)(30,34,36,40);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,31,38)(26,33,35,39)(27,36,37,30)(29,34,32,40);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(25,29,31,32)(26,27,35,37)(28,34,38,40)(30,33,36,39);(1,4,5,3)(2,7,8,6)(11,21)(15,18)(19,20)(23,24)(25,30)(26,34)(27,28)(29,39)(31,36)(32,33)(35,40)(37,38)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=fg,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi

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, 6636048);