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 | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,11)(7,10,8,12)(13,14,18,22,19,15,23,26)(16,24,17,21,27,20,28,25);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,21)(17,24)(18,22)(20,28)(23,26)(25,27);(2,4)(5,8,9,7)(6,10,11,12)(13,16,23,28,19,27,18,17)(14,20,26,21,15,24,22,25);(2,4)(13,17,18,27,19,28,23,16)(14,21,22,20,15,25,26,24);(16,27)(17,28)(20,24)(21,25)Polycyclic presentation:
2^9|a^2=fgi,b^2=gi,c^2=fg,d^2=f,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, 6355329);