#9333318

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

4499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 9240514);