Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,14)(12,15)(13,16)(17,18)(19,20)(21,23)(22,24);(6,8)(9,11)(10,12)(13,15)(14,16)(17,19,21,24)(18,20,23,22);(1,2,3,4)(5,6,7,8)(17,19)(18,22)(20,23)(21,24);(9,11)(10,14)(12,16)(13,15)(17,20,21,22)(18,19,23,24);(9,12,13,14)(10,15,16,11)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=ef128 conjugacy classes — view the full character table.
| maximal subgroup of | #72264575, #72264581, #72264604, … |
| maximal normal subgroup of | #72264575, #72264581, #72264604, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 424992);