Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| 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 | #2428 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1845 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,12)(9,11)(13,14,18,21)(15,23,25,28)(16,22,17,20)(19,26,27,24);(1,2)(3,4)(5,7,9,12)(6,10,11,8)(13,15,16,26,18,25,17,24)(14,19,20,28,21,27,22,23);(5,8,9,10)(6,7,11,12)(13,16,18,17)(14,20,21,22)(15,24,25,26)(19,23,27,28);(2,4)(6,11)(7,12)(14,21)(19,27)(20,22)(23,28);(13,15,16,26,18,25,17,24)(14,19,20,28,21,27,22,23)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7449140);