Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
845 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,24)(16,20)(17,19)(18,23)(21,26)(22,27)(25,28);(1,2,3,4)(13,15,18,25)(14,19,23,21)(16,26,27,17)(20,24,22,28);(2,4)(5,7,8,12)(6,9,10,11)(13,16,18,27)(14,20,23,22)(15,17,25,26)(19,28,21,24);(13,17,16,25,18,26,27,15)(14,21,22,28,23,19,20,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 345896);