Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5020720 |
| automorphism | Aut(G) order 8388608 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×H(3,𝔽4) | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(10,13)(14,15)(17,18)(19,24)(20,23)(21,22);(9,10)(11,14)(12,13)(15,16)(17,19,22,20)(18,23,21,24);(2,3)(9,10,12,13)(11,15,16,14)(17,20,22,19)(18,23,21,24);(1,3,4,2)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16);(9,11,12,16)(10,14,13,15)(17,21,22,18)(19,24,20,23)Polycyclic presentation:
2^9|a^2=gi,b^2=fh,c^2=f,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219213, #72219215, #72219277, … |
| maximal normal subgroup of | #72219213, #72219215, #72219277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4928703);