Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2296 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)(7,9)(8,10)(11,12)(13,14,16,18)(15,22,23,19)(17,20,25,21)(24,27,28,26);(1,2,4,3)(5,7,8,11)(6,9,10,12)(14,18)(19,22)(20,21)(26,27);(6,10)(9,12)(13,15)(14,19)(16,23)(17,24)(18,22)(20,26)(21,27)(25,28);(1,3,4,2)(14,20)(17,25)(18,21)(19,26)(22,27)(24,28);(15,23)(19,22)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=i,b^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,e^c=ei,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536890);