Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1834 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)(13,14,19,24)(15,16,26,27)(17,22,18,23)(20,21,28,25);(5,6)(7,10)(8,9)(11,12)(13,15,18,21,19,26,17,25)(14,20,22,27,24,28,23,16);(2,4)(6,9)(7,11)(13,16,17,28,19,27,18,20)(14,21,23,15,24,25,22,26);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,24,23)(15,25,26,21)(16,20,27,28);(6,9)(10,12)(15,26)(16,27)(20,28)(21,25)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421620);