Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3158 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)(7,10)(8,11)(9,12)(13,14,20,23)(15,24,26,27)(16,21,28,25)(17,18,22,19);(2,4)(5,7,9,8)(6,10,12,11)(13,15)(14,16)(17,24)(18,25)(19,21)(20,26)(22,27)(23,28);(5,7,9,8)(6,11,12,10)(13,16)(14,21)(15,17)(18,24)(19,27)(20,28)(22,26)(23,25);(5,8,9,7)(6,11,12,10)(13,17)(14,18)(15,16)(19,23)(20,22)(21,24)(25,27)(26,28);(13,18,20,19)(14,22,23,17)(15,25,26,21)(16,27,28,24)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360218);