Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437428 |
| 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×#3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,21)(16,20)(17,26)(18,22)(19,24)(23,28)(25,27);(5,7)(6,9)(8,11)(10,12)(13,15,18,25)(14,19,22,28)(16,24,17,23)(20,27,26,21);(1,2,3,4)(6,10)(7,11)(13,16,18,17)(14,20,22,26)(15,23,25,24)(19,21,28,27);(7,11)(9,12)(13,15)(14,21)(16,23)(17,24)(18,25)(19,26)(20,28)(22,27);(13,16,18,17)(14,20,22,26)(15,24,25,23)(19,27,28,21)Polycyclic presentation:
2^9|b^2=i,c^2=gh,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366636);