#8460732

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

4414 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)(3,7)(4,6)(5,8)(9,10,15,14)(11,16,22,21)(12,18,13,17)(19,24,20,23)(27,33)(30,34)(35,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,22)(10,16,14,21)(12,19,13,20)(17,23,18,24)(25,26,29,32)(27,34,33,30)(28,31,37,36)(35,40,39,38);(9,12,15,13)(10,17,14,18)(11,19,22,20)(16,23,21,24)(25,27,29,33)(26,30,32,34)(28,35,37,39)(31,38,36,40);(2,6)(5,7)(9,11)(10,16)(12,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,26)(27,30)(28,36)(29,32)(31,37)(33,34)(35,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,13,15,12)(10,18,14,17)(11,20,22,19)(16,24,21,23);(1,2)(3,5)(4,6)(7,8)(9,14,15,10)(11,21,22,16)(12,18,13,17)(19,24,20,23)(25,28)(26,31)(27,35)(29,37)(30,38)(32,36)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8367928);