Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1786 |
| prime radical | C2×#3150 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,15,11,16,10,28)(7,17,20,35,21,30,29,33)(8,22,25,34,27,36,26,14)(12,23,31,19,13,24,32,18);(1,2)(3,4)(5,7)(6,12)(8,23)(9,20)(10,29)(11,21)(13,16)(14,33)(15,31)(17,22)(18,26)(19,25)(24,27)(28,32)(30,36)(34,35);(2,4)(5,7)(6,13)(8,24)(9,20)(10,29)(11,21)(12,16)(14,33)(15,32)(17,22)(18,25)(19,26)(23,27)(28,31)(30,36)(34,35)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,h^b=hi,d^c=dhi,e^c=eh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#147872, #72215055, #72215063, … |
| maximal normal subgroup of | C2×#147872, #72215055, #72215063, … |
| maximal quotient of | #72214964, C2×#147872 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55068);