Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| 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 | #15947 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,8)(9,10,13,12)(11,15,16,14)(17,18,21,22)(19,23,24,20);(2,4)(6,8)(18,22)(20,23);(1,2,3,4)(5,6,7,8)(18,22)(19,24);(9,11,13,16)(10,14,12,15)(17,19)(18,20)(21,24)(22,23);(9,12,13,10)(11,15,16,14)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=fi,e^2=fi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219260, #72219263, #72219267, … |
| maximal normal subgroup of | #72219260, #72219263, #72219267, … |
| maximal quotient of | #72330502 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 665426);