#8550930

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

4333 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458126);