#10320322

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

4263 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10227518);