Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13523 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2092 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
712 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)(7,17,21,28)(8,13)(9,12)(10,16)(11,18,29,20)(14,23)(15,22)(19,27,24,26)(25,32)(30,34,36,33)(31,35);(1,2,3,4)(5,7,10,21)(6,11)(8,18,22,20)(9,24,25,19)(12,30)(13,17)(14,26)(15,28)(16,29)(23,34,35,33)(27,31)(32,36);(6,12)(7,18)(8,22)(9,23)(11,26)(13,31)(14,15)(16,32)(17,30)(19,24)(20,21)(25,35)(27,29)(28,36)Polycyclic presentation:
2^9|a^2=g,b^2=dh,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,d^c=di,e^c=eh,f^c=fi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239851, #72239860, #72239866, … |
| maximal normal subgroup of | #72239851, #72239860, #72239866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55279);