Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395240 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(10,12)(13,14)(16,20)(17,18)(19,24)(21,23);(1,2,3,4)(9,10,14,17,15,18,13,12)(11,19,23,16,22,20,21,24);(6,8)(9,11)(10,16)(12,20)(13,23)(14,21)(15,22)(17,19)(18,24);(9,12,13,18,15,17,14,10)(11,20,23,24,22,19,21,16)Polycyclic presentation:
2^9|b^2=ei,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^c=ei,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72226904, #72226925, #72226928 |
| maximal normal subgroup of | #72226904, #72226925, #72226928 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319020);