Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535648 |
| 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 | #36259 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,10)(7,12)(8,11)(13,14,17,20)(15,18,21,24)(16,19,23,26)(22,25,27,28);(5,6,11,12,9,7,8,10)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(2,4)(5,7,8,12,9,6,11,10)(15,21)(18,24)(22,27)(25,28);(6,7)(10,12)(13,16)(14,19)(15,22)(17,23)(18,25)(20,26)(21,27)(24,28);(16,23)(19,26)(22,27)(25,28)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,d^c=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6473522);