Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518904 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #59026 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,9,11,12)(13,14)(15,24)(16,21)(17,20)(18,27)(19,23)(22,26)(25,28);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,16,25,19,26,18,17)(14,20,27,22,23,28,21,24);(6,10)(7,11)(13,16,19,18)(14,21,23,27)(15,25,26,17)(20,24,28,22);(2,4)(6,10)(9,12)(13,16,19,18)(14,21,23,27)(15,25,26,17)(20,24,28,22);(13,17,18,26,19,25,16,15)(14,22,21,20,23,24,27,28)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=i,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72308746 |
| maximal normal subgroup of | #72308746 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7434554);