Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| 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×#3054 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,26,17,21,16,22)(15,20,23,19,25,28,24,27);(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,23)(17,25)(18,24)(20,26)(21,27)(22,28);(1,2,3,4)(6,10)(7,11)(13,15)(14,20)(16,24)(17,25)(18,23)(19,26)(21,28)(22,27);(7,11)(9,12)(14,21)(19,27)(20,28)(22,26);(13,16,17,18)(14,22,21,26)(15,23,25,24)(19,28,27,20)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366768);