#8464355

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

4533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=g,e^2=hi,f^2=gi,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, 8371551);