Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,21)(22,23);(6,8)(10,12)(14,15)(17,19)(18,21)(20,22)(23,24);(5,6,7,8)(11,16)(14,15)(18,23)(19,20);(1,2)(3,4)(9,11)(10,14)(12,15)(13,16)(17,20,22,19)(18,21,23,24);(9,12,13,10)(11,14,16,15)(17,21)(18,20)(19,23)(22,24)Polycyclic presentation:
2^9|c^2=gh,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232467, #72232508, #72232528, … |
| maximal normal subgroup of | #72232467, #72232508, #72232528, … |
| maximal quotient of | #72328926, #72330462, #72331316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2978753);