#8505268

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
prime radical	#39426	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4221 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)(3,4)(5,6)(7,8)(9,10,13,14)(11,17,19,15)(12,16,21,22)(18,24,20,23)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38);(2,6)(3,7)(10,14)(15,17)(16,22)(23,24)(27,33)(30,37)(34,39)(38,40);(1,3,5,7)(2,4,6,8)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,4)(2,7)(3,6)(5,8)(9,11,13,19)(10,15,14,17)(12,20,21,18)(16,24,22,23)(25,27,29,33)(26,30,32,37)(28,34,36,39)(31,38,35,40);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40);(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8412464);