#8901650

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

4351 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=gi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8808846);