Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(7,11)(10,12)(13,14,18,21,19,22,17,27)(15,23,25,16,24,26,28,20);(5,6)(7,10)(8,9)(11,12)(13,15,19,24)(14,20,22,16)(17,25,18,28)(21,26,27,23);(1,2,3,4)(6,9)(10,12)(13,16,19,20)(14,15,22,24)(17,23,18,26)(21,25,27,28);(6,9)(10,12)(15,24)(16,20)(23,26)(25,28);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,22,27)(15,25,24,28)(16,23,20,26)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6401372);