Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,7)(13,14,17,21,18,22,16,19)(15,23,24,20,26,27,25,28);(9,10,11,12)(14,19)(16,17)(20,27)(21,22)(23,28)(24,25);(2,4)(13,15,18,26)(14,20,22,28)(16,24,17,25)(19,27,21,23);(5,7,8,6)(13,16,18,17)(14,21,22,19)(15,24,26,25)(20,23,28,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261636);