Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433049 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(7,11)(10,12)(13,14,19,21)(15,17,22,18)(16,24,27,20)(23,26,25,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,14,19,22,17,21)(16,25,26,24,27,23,28,20);(2,4)(6,9)(10,12)(13,16)(14,20)(15,23)(17,28)(18,26)(19,27)(21,24)(22,25);(5,7,8,11)(6,10,9,12)(14,21)(15,22)(20,24)(23,25);(13,17,19,18)(14,22,21,15)(16,26,27,28)(20,23,24,25)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,d^b=di,e^b=ei,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342767);