Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394590 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,27,11,15,18,16)(7,17,20,33,21,12,28,35)(8,22,29,32,24,36,19,13)(9,25,30,31,23,14,26,34);(1,2)(3,4)(5,7,18,20,11,21,10,28)(6,12,16,35,15,17,27,33)(8,23,29,30,24,9,19,26)(13,34,22,14,32,31,36,25);(2,4)(5,8,10,29,11,24,18,19)(6,13,27,22,15,32,16,36)(7,9,20,30,21,23,28,26)(12,31,35,14,17,34,33,25);(5,9)(6,14)(7,19)(8,20)(10,30)(11,23)(12,32)(13,17)(15,25)(16,31)(18,26)(21,29)(22,33)(24,28)(27,34)(35,36)Polycyclic presentation:
2^9|a^2=ef,b^2=efh,c^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 314462);