Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3054 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(10,14)(12,15)(16,20)(17,21)(19,22)(23,24);(9,10,15,20,13,16,12,14)(11,17,22,24,18,23,19,21);(2,4)(10,16)(14,20)(17,23)(21,24);(5,7,8,6)(9,11)(10,17)(12,19)(13,18)(14,21)(15,22)(16,23)(20,24);(11,18)(17,23)(19,22)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280185);