#8460730

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

4343 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,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, 8367926);