Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2190 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,15)(8,20)(9,24)(10,26)(11,16)(12,19)(13,17)(14,25)(18,33)(21,32)(22,28)(23,36)(27,29)(30,34)(31,35);(1,2,3,4)(5,7,9,19,11,18,25,10)(6,12,14,15,16,26,24,33)(8,17,21,35,22,34,23,27)(13,28,29,32,30,20,31,36);(5,8)(6,13)(7,17)(9,21)(10,27)(11,22)(12,28)(14,29)(15,32)(16,30)(18,34)(19,35)(20,26)(23,25)(24,31)(33,36);(7,18)(10,19)(12,26)(15,33)(17,34)(20,28)(27,35)(32,36)Polycyclic presentation:
2^9|b^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^c=fh,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250058, #72250084, #72250096 |
| maximal normal subgroup of | #72250058, #72250084, #72250096 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419092);