#9730423

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

4469 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9637619);