Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2840 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)(9,10)(11,17)(12,21)(13,23)(14,16)(15,20)(18,22)(19,24);(1,2)(3,4)(5,6)(7,8)(9,11,13,20,14,19,22,12)(10,15,18,17,16,21,23,24);(6,8)(10,16)(11,19)(12,20)(18,23);(2,4)(6,8)(10,16)(15,21)(17,24)(18,23);(9,12,22,19,14,20,13,11)(10,17,23,15,16,24,18,21)Polycyclic presentation:
2^9|b^2=f,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308113, #72308908 |
| maximal normal subgroup of | #72308113, #72308908 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426154);