Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,8)(10,11)(13,14,17,21,18,22,26,27)(15,23,24,20,25,28,16,19);(9,10)(11,12)(13,15,18,25)(14,19,22,20)(16,26,24,17)(21,28,27,23);(1,2,3,4)(9,10,12,11);(6,8)(9,11,12,10)(13,16,17,15,18,24,26,25)(14,20,21,28,22,19,27,23)Polycyclic presentation:
2^9|a^2=eg,b^2=i,c^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341411);