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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3056 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)(13,14,18,24,20,16,19,23)(15,17,22,27,26,28,25,21);(2,4)(6,10)(8,11)(13,15)(14,21)(16,27)(17,23)(18,25)(19,22)(20,26)(24,28);(5,6)(7,8)(9,10)(11,12)(13,16,20,14)(15,21,26,27)(17,22,28,25)(18,24,19,23);(5,7,9,12)(6,11,10,8)(13,17,20,28)(14,22,16,25)(15,24,26,23)(18,27,19,21);(5,8,9,11)(6,7,10,12)(13,18,20,19)(14,23,16,24)(15,22,26,25)(17,21,28,27)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,e^2=h,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, 6524638);