Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
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 | #1999 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
749 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)(5,6,11,7,10,9,8,12)(14,19,21,26)(20,23,27,28);(5,7,10,12)(6,8,9,11)(13,14,15,19,18,21,17,26)(16,23,22,27,25,28,24,20);(2,4)(5,8,10,11)(6,7,9,12)(13,15,18,17)(14,19,21,26)(16,22,25,24)(20,23,27,28);(5,9,8,7,10,6,11,12)(13,16,17,24,18,25,15,22)(14,20,19,23,21,27,26,28)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=h,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=eh,g^b=gh,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 137144);