#8422101

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

4431 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8329297);