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 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3349 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)(9,10,14,18)(11,15,20,13)(12,17,23,16)(19,24,21,22);(9,11)(10,15)(12,21,23,19)(13,18)(14,20)(16,22,17,24);(2,5)(6,8)(9,12)(10,16)(11,19)(13,24)(14,23)(15,22)(17,18)(20,21);(9,12)(10,17)(11,19)(13,22)(14,23)(15,24)(16,18)(20,21);(1,3,4,7)(2,6,5,8)(9,13,14,15)(10,11,18,20)(12,22,23,24)(16,21,17,19)Polycyclic presentation:
2^9|a^2=i,b^2=f,e^2=g,b^a=bf,c^a=cg,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 | #72318286, #72318292, #72318314, … |
| maximal normal subgroup of | #72318286, #72318292, #72318314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6362078);