#9241172

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

4518 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,e^2=g,f^2=g,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, 9148368);