Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(7,11)(10,12)(13,14,18,26,17,19,16,22)(15,21,23,20,24,28,25,27);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,19)(20,27)(21,28)(22,26);(6,9)(10,12)(13,15)(14,20)(16,23)(17,24)(18,25)(19,27)(21,26)(22,28);(5,7,8,11)(6,10,9,12)(13,15)(14,21)(16,25)(17,24)(18,23)(19,28)(20,26)(22,27);(13,16,17,18)(14,22,19,26)(15,23,24,25)(20,28,27,21)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342716);