Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135930 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,17,20,28)(8,22)(9,13)(10,27)(11,16)(12,18,30,34)(14,24)(15,26)(19,35,25,31)(21,33,36,29)(23,32);(5,7,8,18,10,21,9,25,11,20,24,34,26,36,23,19)(6,12,13,17,15,31,22,29,16,30,32,28,27,35,14,33);(2,4)(7,18)(8,23)(9,24)(10,26)(12,28)(13,14)(15,27)(17,30)(19,21)(20,34)(22,32)(25,36)(29,31)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=dhi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224903, #72224931, #72226810, … |
| maximal normal subgroup of | #72224903, #72224931, #72226810, … |
| maximal quotient of | #72224895, #72237230, C2×#148376 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55572);