Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535623 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #28397 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2178 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)(5,6,11,10)(7,8,9,12)(13,14,18,20)(15,22,25,28)(16,21,17,26)(19,23,27,24);(5,7,11,9)(6,12,10,8)(13,15,17,24,18,25,16,23)(14,19,21,22,20,27,26,28);(1,2,3,4)(5,8,11,12)(6,7,10,9)(14,20)(19,27)(21,26)(22,28);(5,9,11,7)(6,12,10,8)(13,16,18,17)(14,21,20,26)(15,23,25,24)(19,22,27,28);(5,10,11,6)(7,8,9,12)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=gi,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^d=eh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6477732);