Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| 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 | C2×#2408 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(9,10)(11,15)(12,14)(13,16)(18,21)(20,23);(1,2,4,3)(6,8)(9,10,13,16)(11,14,12,15)(17,18,22,21)(19,23,24,20);(5,6,7,8)(9,11,13,12)(10,14,16,15)(17,19,22,24)(18,23,21,20);(9,10)(11,14)(12,15)(13,16)(17,20)(18,19)(21,24)(22,23);(1,3,4,2)(9,12,13,11)(10,15,16,14)(17,21,22,18)(19,20,24,23)Polycyclic presentation:
2^9|b^2=f,c^2=ghi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72263606, #72263610, #72263626 |
| maximal normal subgroup of | #72263606, #72263610, #72263626 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 763763);