#10361083

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

4474 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10268279);