#8509524

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416720);