#9189522

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

4504 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=g,e^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, 9096718);