Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129430 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C23×C4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1799 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
675 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,17)(8,12)(9,22)(10,25,27,15)(11,16)(13,20,32,26)(14,24)(18,34,35,31)(19,33)(21,29)(23,30,36,28);(1,2)(3,4)(5,7,9,8,11,19,24,21)(6,12,14,17,16,29,22,33)(10,18,23,20,27,35,36,26)(13,30,31,25,32,28,34,15);(2,4)(5,8,9,19,11,21,24,7)(6,13,14,31,16,32,22,34)(10,26,23,18,27,20,36,35)(12,28,17,15,29,30,33,25)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,e^c=ei,g^c=gh,g^e=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55897);