Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533141 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(12,13)(15,22)(16,17)(19,20);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22);(2,5)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,24)(18,22);(1,3)(2,6)(4,7)(5,8)(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,23,22,24);(3,7)(6,8)(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,24,22,23)Polycyclic presentation:
2^9|d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,g^b=gi,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6440339);