Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
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 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10)(11,21)(12,18)(13,19)(14,22)(15,20)(16,17)(23,24);(5,6)(7,8)(9,11,14,17,15,23,13,18)(10,16,19,21,20,12,22,24);(1,2,3,4)(6,8)(9,12)(10,17)(11,22)(13,21)(14,24)(15,16)(18,20)(19,23);(6,8)(9,12)(10,18)(11,19)(13,21)(14,24)(15,16)(17,20)(22,23);(9,13,15,14)(10,19,20,22)(11,18,23,17)(12,24,16,21)Polycyclic presentation:
2^9|b^2=f,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350330);