Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1119 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)(5,6,8,10,7,9,11,12)(14,18)(16,25)(19,26)(20,21)(22,27)(23,28);(6,9)(10,12)(13,14,17,21)(15,22,24,19)(16,20,25,18)(23,26,28,27);(2,4)(13,15,17,24)(14,19,21,22)(16,23,25,28)(18,26,20,27);(6,9)(10,12)(14,20)(16,25)(18,21)(19,27)(22,26)(23,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ch,d^b=dei,e^b=eh,e^d=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 127336);