#8425437

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

4608 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=gh,e^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, 8332633);