#8926593

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

4353 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833789);