#9030140

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

4356 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)(3,4)(5,6)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,26,31,29)(27,28,35,32)(30,37,38,33)(34,39,40,36);(2,6)(4,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,22)(21,24)(25,27,31,35)(26,32,29,28)(30,34,38,40)(33,39,37,36);(1,2,5,6)(3,7,8,4)(10,14)(11,13)(12,20)(23,24)(26,29)(28,32)(30,38)(34,40);(1,3)(2,7)(4,6)(5,8)(9,12,15,20)(10,17,14,21)(11,19,13,22)(16,23,18,24)(25,28)(26,27)(29,35)(30,36)(31,32)(33,34)(37,40)(38,39);(1,4,5,7)(2,3,6,8)(9,13,15,11)(10,18,14,16)(12,19,20,22)(17,23,21,24)(25,29,31,26)(27,32,35,28)(30,37,38,33)(34,39,40,36);(1,3)(2,7)(4,6)(5,8)(9,14,15,10)(11,18,13,16)(12,21,20,17)(19,24,22,23)(25,30,31,38)(26,33,29,37)(27,34,35,40)(28,36,32,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8937336);