Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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×#3062 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14,18,24,19,23,20,16)(15,25,26,22,27,28,17,21);(6,9)(10,12)(13,15)(14,21)(16,25)(17,18)(19,27)(20,26)(22,23)(24,28);(1,2,3,4)(13,16,18,14,19,24,20,23)(15,22,26,28,27,21,17,25);(5,7,8,11)(6,10,9,12)(13,17,18,15,19,26,20,27)(14,22,24,28,23,21,16,25);(14,23)(16,24)(21,22)(25,28)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367213);