Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #38050 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2956 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)(13,14,17,20,18,19,27,25)(15,21,23,22,24,28,26,16);(2,4)(5,6,8,9)(7,10,11,12)(13,14)(15,22)(16,24)(17,25)(18,19)(20,27)(21,23)(26,28);(6,9)(7,11)(13,15,18,24)(14,16,19,22)(17,26,27,23)(20,28,25,21);(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,21)(18,22)(19,24)(20,23)(25,26)(27,28);(6,9)(7,11)(14,19)(16,22)(20,25)(21,28)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503716);