#9063999

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

4389 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8971195);