Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| 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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(13,14,17,21,18,22,26,19)(15,23,24,27,16,20,25,28);(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(14,19)(15,16)(17,26)(20,28)(21,22)(23,27);(1,3,4,2)(6,9)(10,11)(13,15,18,16)(14,20,22,23)(17,24,26,25)(19,27,21,28);(5,7,8,12)(6,10,9,11)(13,16,18,15)(14,20,22,23)(17,25,26,24)(19,27,21,28);(15,16)(20,23)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=fgi,b^2=gi,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371306);