Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(9,10,14,11)(12,20,23,21)(13,18,15,16)(17,24,19,22);(1,2,4,3)(5,6)(7,8)(9,11)(10,14)(12,21)(13,16)(15,18)(17,24)(19,22)(20,23);(10,16)(11,18)(13,15)(17,20)(19,21)(22,24);(6,8)(9,12,14,23)(10,17,11,19)(13,22,15,24)(16,20,18,21);(1,3,4,2)(9,13,14,15)(10,16,11,18)(12,22,23,24)(17,20,19,21)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72315341, #72315342, #72315348 |
| maximal normal subgroup of | #72315341, #72315342, #72315348 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426449);