Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470428 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2149 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,19)(15,22,23,28)(16,20,26,25)(17,24,27,21);(7,8)(10,12)(13,15,17,25,18,23,27,20)(14,16,21,22,19,26,24,28);(1,2,3,4)(5,7,9,8)(6,10,11,12)(14,19)(15,23)(20,25)(21,24);(5,8,9,7)(6,10,11,12)(13,16,18,26)(14,20,19,25)(15,24,23,21)(17,22,27,28);(6,11)(10,12)(14,19)(16,26)(21,24)(22,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390555);