Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10)(11,12)(13,14)(15,16)(18,22)(23,24);(10,14)(12,15)(17,18,21,22)(19,24,20,23);(5,6)(7,8)(17,19,21,20)(18,23,22,24);(1,2)(3,4)(5,6)(7,8)(9,11)(10,15)(12,14)(13,16);(9,12,13,15)(10,11,14,16)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^9|b^2=h,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72311428, #72311453, #72311550, … |
| maximal normal subgroup of | #72311428, #72311453, #72311550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 556978);