Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1302 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)(7,16)(8,19)(9,14)(10,25)(11,12)(13,23)(15,21)(17,31)(18,30)(20,34)(22,33)(24,27)(26,35)(28,36)(29,32);(1,2,4,3)(5,7,9,11)(6,12,14,16)(8,20,22,17)(10,21,24,18)(13,26,28,29)(15,25,30,27)(19,31,33,34)(23,32,36,35);(5,8,21,35,11,20,24,36,9,22,18,32,7,17,10,23)(6,13,25,31,16,29,30,33,14,28,27,34,12,26,15,19);(1,3,4,2)(6,14)(12,16)(13,28)(15,30)(19,33)(25,27)(26,29)(31,34)Polycyclic presentation:
2^9|b^2=e,c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72224877 |
| maximal normal subgroup of | #72224877 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410316);