Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533105 |
| 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 |
| prime radical | #36455 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3152 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)(6,11)(7,12)(8,9);(2,4)(5,6,10,7)(8,11,9,12)(13,14)(15,19)(16,20)(17,18)(21,27)(22,25)(23,28)(24,26);(5,7,10,6)(8,12,9,11)(14,18)(19,25)(20,26)(27,28);(5,8,10,9)(6,11,7,12)(13,15,17,22)(14,19,18,25)(16,23,24,21)(20,28,26,27);(13,16,17,24)(14,20,18,26)(15,21,22,23)(19,27,25,28)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,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, 6451279);