Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394048 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,9)(7,10,11,12)(14,15)(16,23,24,26)(19,27,21,28)(20,22);(1,2)(3,4)(6,9)(10,12)(13,14,17,15)(16,19,24,21)(18,22,25,20)(23,27,26,28);(2,4)(13,15,17,14)(16,21,24,19)(18,20,25,22)(23,28,26,27);(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,21)(17,24)(18,26)(20,27)(22,28)(23,25)Polycyclic presentation:
2^9|a^2=ei,b^2=i,c^2=i,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307322);