Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394020 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1739 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,12)(8,19)(9,24)(10,14)(11,15)(13,21)(16,33)(17,28)(18,29)(20,26)(22,31)(23,32)(25,27)(30,34)(35,36);(1,2,4,3)(5,7,11,18)(6,12,15,29)(8,20,23,35)(9,17,10,25)(13,30,31,33)(14,27,24,28)(16,21,34,22)(19,26,32,36);(5,8)(6,13)(7,16)(9,22)(10,21)(11,23)(12,26)(14,32)(15,31)(17,35)(18,34)(19,24)(20,25)(27,30)(28,33)(29,36);(1,3,4,2)(5,6)(7,12)(8,13)(9,24)(10,14)(11,15)(16,26)(17,28)(18,29)(19,22)(20,30)(21,32)(23,31)(25,27)(33,35)(34,36)Polycyclic presentation:
2^9|b^2=ei,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302418);