Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6,8,7)(13,14)(15,22)(16,20)(17,19)(18,23)(21,25)(24,26)(27,28);(6,7)(9,10,11,12)(13,15)(14,18)(16,23)(17,24)(19,27)(20,22)(21,26)(25,28);(14,19)(15,23)(17,25)(18,26)(20,21)(22,27);(5,7,8,6)(15,23)(18,22)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=ei,f^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=de,f^c=fi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263391);