Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,12)(9,11)(13,14,18,22,17,19,24,28)(15,20,25,27,23,26,16,21);(2,3)(5,7,9,8)(6,10,11,12)(14,19)(15,23)(16,25)(22,28);(5,8,9,7)(6,10,11,12)(13,15)(14,20)(16,24)(17,23)(18,25)(19,26)(21,28)(22,27);(1,3,4,2)(13,16,17,25)(14,21,19,27)(15,24,23,18)(20,28,26,22);(15,23)(16,25)(20,26)(21,27)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ci,d^b=df,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287881);