Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1799 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,18)(8,21)(9,24,26,14)(10,16)(11,17)(12,19,30,34)(13,20)(15,27)(22,29,35,31)(23,33)(25,32,36,28);(5,7,10,8,11,20,27,23)(6,12,16,31,17,30,15,29)(9,19,25,22,26,34,36,35)(13,32,33,14,18,28,21,24);(1,2,3,4)(5,8,10,20,11,23,27,7)(6,13,15,21,17,18,16,33)(9,22,25,34,26,35,36,19)(12,28,29,14,30,32,31,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=ci,f^b=fh,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279553, #72279591, #72279633, … |
| maximal normal subgroup of | #72279553, #72279591, #72279633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54219);