Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574015 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2221 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)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,18,20)(15,23,24,26)(16,21,19,28)(17,25,27,22);(5,7)(6,9)(8,12)(10,11)(13,15,17,21,18,24,27,28)(14,19,22,23,20,16,25,26);(2,4)(7,12)(9,10)(14,20)(15,24)(21,28)(22,25);(5,7)(6,10)(8,12)(9,11)(13,16)(14,21)(15,25)(17,26)(18,19)(20,28)(22,24)(23,27);(7,12)(9,10)(15,24)(16,19)(21,28)(23,26)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6504252);