Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3257 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2037 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22)(23,30)(24,35)(25,31)(26,32)(27,36)(28,33)(29,34);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,27,34,28,31,26,24)(22,29,32,30,33,35,36,25);(7,13)(8,15)(10,16)(11,18)(21,24,27,23,28,34,26,31)(22,30,32,35,33,25,36,29);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(21,25,28,30)(22,31,33,23)(24,36,34,32)(26,35,27,29);(7,13)(10,16)(14,19)(17,20)(21,26,28,27)(22,32,33,36)(23,24,31,34)(25,29,30,35)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6434268);