Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| 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 | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6)(7,8)(10,11)(12,16)(18,22)(23,24);(2,4)(6,8)(10,11)(14,15)(18,22)(23,24);(1,2)(3,4)(5,6)(7,8)(9,10,13,11)(12,14,16,15)(17,18)(19,23)(20,24)(21,22);(9,11,13,10)(12,15,16,14)(17,19,21,20)(18,23,22,24);(9,12,13,16)(10,14,11,15)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^9|c^2=i,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72305087, #72305130, #72305135 |
| maximal normal subgroup of | #72305087, #72305130, #72305135 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5863145);