Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574015 |
| 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 | #2856 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1867 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)(5,6,9,8)(7,10,12,11)(13,14,18,22)(15,23,25,26)(16,20,19,28)(17,24,27,21);(6,8)(10,11)(13,15,17,20,18,25,27,28)(14,19,21,23,22,16,24,26);(2,4)(5,6,9,8)(7,11,12,10)(13,16)(14,20)(15,24)(17,26)(18,19)(21,25)(22,28)(23,27);(5,7,9,12)(6,10,8,11)(15,25)(16,19)(20,28)(23,26);(5,8,9,6)(7,10,12,11)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=h,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6534546);