Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,19,22,18,17,21,24)(15,23,26,27,25,16,20,28);(1,2,4,3)(6,9)(10,12)(13,15)(14,16)(17,23)(18,25)(19,26)(20,21)(22,28)(24,27);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16,18,23)(14,20,17,26)(15,24,25,22)(19,28,21,27);(13,17,19,24,18,14,21,22)(15,16,26,28,25,23,20,27)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72317237, #72317284, #72317286 |
| maximal normal subgroup of | #72317237, #72317284, #72317286 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266445);