Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 | |
3310 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)(11,21)(12,20)(13,24)(14,18)(15,17)(16,22)(19,23);(1,2)(3,4)(5,6)(7,8)(9,11,15,13)(10,16,17,19)(12,22,14,23)(18,21,20,24);(6,8)(10,17)(16,19)(18,20)(21,24);(2,4)(6,8)(9,12,15,14)(10,18,17,20)(11,22,13,23)(16,21,19,24);(9,13,15,11)(10,19,17,16)(12,23,14,22)(18,24,20,21)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,f^e=fi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421403);