#8436553

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

4347 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=gh,e^2=g,f^2=g,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, 8343749);