Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #269019 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#1600 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1403 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,17,20)(15,22,23,19)(16,24,25,28)(18,27,26,21);(1,2,4,3)(5,7)(6,9)(8,12)(10,11)(13,15,17,23)(14,19,20,22)(16,21,25,27)(18,24,26,28);(5,6)(7,11,12,9)(8,10)(13,16,18,19,17,25,26,22)(14,15,21,24,20,23,27,28);(1,3,4,2)(14,20)(15,23)(21,27)(24,28)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 204516);