Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1885 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14,17,21,18,22,19,23)(15,24,25,28,26,27,16,20);(1,2)(3,4)(13,15)(14,20)(16,17)(18,26)(19,25)(21,27)(22,28)(23,24);(2,4)(5,7,8,12)(6,9,10,11)(13,14,19,23,18,22,17,21)(15,24,16,20,26,27,25,28);(2,4)(13,16,17,15,18,25,19,26)(14,20,21,24,22,28,23,27);(13,17,18,19)(14,21,22,23)(15,16,26,25)(20,27,28,24)Polycyclic presentation:
2^9|a^2=fgi,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353396);