#8182909

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

4509 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8090105);