Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994107 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2169 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,17)(15,23,27,25)(16,22,18,21)(20,26,28,24);(5,6)(7,11)(8,10)(9,12)(13,15,19,27)(14,20,17,28)(16,24,18,26)(21,23,22,25);(2,4)(5,7,8,12)(6,9,10,11)(13,16,19,18)(14,21,17,22)(15,24,27,26)(20,23,28,25);(7,12)(9,11)(13,17,19,14)(15,25,27,23)(16,21,18,22)(20,24,28,26);(6,10)(9,11)(13,16,19,18)(14,22,17,21)(15,26,27,24)(20,23,28,25)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dhi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6962070);