Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1874 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,19)(9,24)(10,13)(11,16)(12,23)(14,22)(17,31)(18,28)(20,27)(21,36)(25,32)(26,34)(29,33)(30,35);(5,7)(6,11,13,26)(8,20)(9,25)(10,17)(12,28,29,32)(14,30,24,27)(15,19,31,36)(16,33)(18,22)(21,35)(23,34);(1,3,4,2)(8,21)(12,29)(15,31)(16,34)(19,36)(20,35)(23,33)(28,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dhi,e^b=ei,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54420);