Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,8)(7,10,12,11)(13,14,18,21)(15,17,20,24)(16,22,26,28)(19,27,23,25);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15)(14,17)(16,23)(18,20)(19,26)(21,24)(22,25)(27,28);(5,8,9,6)(7,11,12,10)(13,16,24,27,18,26,17,25)(14,19,20,28,21,23,15,22);(7,12)(10,11)(13,16,17,25,18,26,24,27)(14,19,15,22,21,23,20,28);(1,3,4,2)(13,17,18,24)(14,20,21,15)(16,25,26,27)(19,28,23,22)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=hi,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511403);