Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2065 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,18,21,34)(8,23)(9,26)(10,29)(11,16)(12,19,31,35)(13,20,32,27)(14,24)(15,25)(17,28)(22,33,36,30);(5,7,8,19,10,22,24,20,11,21,25,35,28,36,9,27)(6,12,14,18,17,32,23,30,16,31,26,34,29,13,15,33);(2,4)(5,7,8,19,10,22,24,20,11,21,25,35,28,36,9,27)(6,13,14,33,17,12,23,18,16,32,26,30,29,31,15,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,e^c=ei,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56077);