#8935141

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

4418 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842337);