#8504963

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

4521 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=i,e^2=g,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, 8412159);