#8633730

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

4376 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8540926);