Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #2541 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#554 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10)(11,18)(12,16)(13,17,21,15)(14,19)(20,23,24,22);(1,2,5,6)(3,7,4,8)(9,11,12,19)(10,14,16,18)(13,20,21,24)(15,23,17,22);(1,3,5,4)(2,7,6,8)(10,15)(11,19)(13,21)(14,22)(16,17)(18,23);(1,4,5,3)(2,7,6,8)(11,19)(14,18)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=ei,d^2=ei,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fi,g^c=ghi,g^f=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404849);