Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2420 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1646 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)(5,6,12,17)(7,18,21,15)(8,22)(9,13,20,29)(10,26,11,16)(14,25)(19,30)(23,36)(24,31)(27,32)(28,35)(33,34);(1,2,4,3)(5,7,11,20,12,21,10,9)(6,13,26,15,17,29,16,18)(8,23,24,34,25,35,27,19)(14,30,31,28,22,33,32,36);(5,8)(6,14)(7,19)(9,23)(10,24)(11,27)(12,25)(13,28)(15,30)(16,31)(17,22)(18,33)(20,35)(21,34)(26,32)(29,36);(1,3,4,2)(5,9,10,21,12,20,11,7)(6,15,16,13,17,18,26,29)(8,23,24,34,25,35,27,19)(14,30,31,28,22,33,32,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272245);