#8183772

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

4513 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)(4,6)(7,8)(9,10,15,14)(11,18,13,16)(12,20,22,17)(19,23,21,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,39)(34,37)(38,40);(2,3)(6,7)(9,11,15,13)(10,16,14,18)(12,19,22,21)(17,23,20,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40);(4,8)(6,7)(10,14)(16,18)(17,20)(23,24)(26,30)(31,37)(32,35)(38,39);(1,3,5,2)(4,7,8,6)(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,27,29,34)(26,31,30,37)(28,33,36,40)(32,38,35,39);(9,13,15,11)(10,18,14,16)(12,21,22,19)(17,24,20,23)(25,28,29,36)(26,32,30,35)(27,33,34,40)(31,38,37,39);(1,4,5,8)(2,6,3,7)(9,14,15,10)(11,18,13,16)(12,20,22,17)(19,24,21,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8090968);