Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2757 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,23,24,19)(16,20,25,28)(17,21,26,27);(1,2,4,3)(6,9)(10,12)(13,15)(14,19)(16,20)(17,21)(18,24)(22,23)(25,28)(26,27);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16,18,25)(14,20,22,28)(15,17,24,26)(19,27,23,21);(13,17,18,26)(14,21,22,27)(15,16,24,25)(19,28,23,20);(14,22)(15,24)(16,25)(21,27)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347911);