Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2896 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)(7,9)(8,10)(11,12)(13,14,17,23,18,20,16,22)(15,21,24,27,25,28,26,19);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,27)(21,22)(23,28);(2,4)(6,10)(9,12)(14,20)(19,27)(21,28)(22,23);(2,4)(13,15)(14,21)(16,26)(17,24)(18,25)(19,22)(20,28)(23,27);(13,16,18,17)(14,22,20,23)(15,24,25,26)(19,21,27,28)Polycyclic presentation:
2^9|a^2=f,b^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347811);