Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437452 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3251 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,18)(13,21)(14,16)(15,19)(20,23)(22,24);(9,11,14,20)(10,15,16,24)(12,19,13,22)(17,18,23,21);(2,5)(6,8)(10,16)(12,13)(17,23)(19,22);(9,11)(10,17)(12,19)(13,22)(14,20)(15,18)(16,23)(21,24);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,18,16,21)(11,19,20,22)(15,23,24,17)Polycyclic presentation:
2^9|b^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,f^b=fi,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318145, #72318164, #72318196, … |
| maximal normal subgroup of | #72318145, #72318164, #72318196, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361058);