#8115528

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

4266 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)(3,7,5,8)(9,10)(11,17)(12,20)(13,16)(14,18)(15,21)(19,22)(23,24)(25,26)(27,32)(28,33)(29,31)(30,34)(35,39)(36,40)(37,38);(1,3,6,5)(2,7,4,8)(9,10,13,16)(11,14,18,17)(12,15,21,20)(19,22,24,23)(25,27,31,30)(26,32,29,34)(28,36,38,35)(33,40,37,39);(1,3,6,5)(2,8,4,7)(11,18)(14,17)(19,24)(22,23)(25,26,31,29)(27,34,30,32)(28,37,38,33)(35,39,36,40);(1,4,6,2)(3,7,5,8)(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(25,28,31,38)(26,33,29,37)(27,35,30,36)(32,39,34,40);(1,5,6,3)(2,7,4,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,29,31,26)(27,32,30,34)(28,37,38,33)(35,39,36,40);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,30,31,27)(26,32,29,34)(28,36,38,35)(33,39,37,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8022724);