#9189493

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

4439 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=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, 9096689);