Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437999 |
| 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 | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3644 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,24)(20,23)(21,22);(9,11)(10,15)(12,20,21,17)(13,22,23,16)(14,19)(18,24);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,17)(13,15)(14,21)(18,22)(19,20)(23,24);(1,4,5,3)(2,6,7,8)(9,13)(10,17)(11,16)(12,15)(14,23)(18,20)(19,22)(21,24);(10,18)(12,21)(15,24)(17,20)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=g,b^a=bf,c^a=cg,e^a=ei,h^a=hi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318717, #72319001 |
| maximal normal subgroup of | #72318717, #72319001 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359470);