Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2456 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3160 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)(5,6,8,7)(10,14)(12,16)(17,18,23,24)(19,22,21,20);(9,10,13,14)(11,12,15,16)(17,19,23,21)(18,20,24,22);(1,2)(3,4)(5,7,8,6)(10,14)(11,15)(17,20,23,22)(18,19,24,21);(1,2)(3,4)(9,11)(10,12)(13,15)(14,16)(17,21,23,19)(18,20,24,22);(9,12)(10,11)(13,16)(14,15)(17,22,23,20)(18,19,24,21)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df116 conjugacy classes — view the full character table.
| maximal subgroup of | #72282109, #72282204, #72282240, … |
| maximal normal subgroup of | #72282109, #72282204, #72282240, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 444636);