Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394797 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1613 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,20)(9,24)(10,14)(11,15)(12,18)(13,23)(17,33)(19,28)(21,27)(22,31)(25,29)(26,32)(30,35)(34,36);(5,7,9,18,11,19,10,25)(6,12,14,16,15,29,24,28)(8,21,22,17,23,34,26,35)(13,30,31,36,20,33,32,27);(5,8,9,22,11,23,10,26)(6,13,14,31,15,20,24,32)(7,17,18,34,19,35,25,21)(12,27,16,30,29,36,28,33);(2,4)Polycyclic presentation:
2^9|b^2=e,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301993);