Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| 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 | C2×#3035 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,11)(7,10,8,12)(13,14,18,23,20,25,19,24)(15,26,16,22,27,28,17,21);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,21)(16,19)(17,18)(20,27)(22,25)(23,28)(24,26);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,27,20,17,19,15)(14,22,23,28,25,21,24,26);(2,4)(13,17,18,15,20,16,19,27)(14,21,23,26,25,22,24,28);(13,18,20,19)(14,23,25,24)(15,17,27,16)(21,28,22,26)Polycyclic presentation:
2^9|a^2=fg,b^2=g,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^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355332);