Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #381 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,18,20,15)(12,14)(13,17)(16,21)(19,23,24,22);(5,6)(7,8)(10,14,17,16)(15,22,18,23);(2,4)(6,8)(9,11)(10,15)(12,19)(13,20)(14,22)(16,23)(17,18)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,h^a=hi,c^b=cf,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72235491, #72260577, #72260596, … |
| maximal normal subgroup of | #72235491, #72260577, #72260596, … |
| maximal quotient of | #72235541, #72261167 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30713);