#10038544

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

4364 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9945740);