Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4839 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,15,20)(11,18,13,23)(12,17,21,16)(14,24,22,19);(5,6)(7,8)(9,11,15,13)(10,16,20,17)(12,22,21,14)(18,19,23,24);(2,4)(6,8)(9,12)(10,17)(11,14)(13,22)(15,21)(16,20)(18,24)(19,23);(5,6)(7,8)(9,13,15,11)(10,18,20,23)(12,22,21,14)(16,19,17,24);(6,8)(11,13)(12,21)(16,17)(18,23)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72281696, #72281760, #72282243, … |
| maximal normal subgroup of | #72281696, #72281760, #72282243, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480826);