Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3488 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4352 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)(3,7)(4,6)(5,8)(9,10,13,19,14,18,12,17)(11,15,20,23,21,16,22,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,22)(13,20)(14,21)(16,18)(17,24)(19,23);(9,11)(10,16)(12,20)(13,22)(14,21)(15,18)(17,24)(19,23);(3,8)(5,7);(2,6)(5,7)(9,12,14,13)(10,17,18,19)(11,20,21,22)(15,23,16,24)Polycyclic presentation:
2^9|a^2=h,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72303360 |
| maximal normal subgroup of | #72303360 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293190);