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 | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(5,6,7,8)(9,10,13,14)(11,16,15,12)(18,22)(20,24);(10,14)(12,16)(17,18)(19,20)(21,22)(23,24);(1,2,4,3)(10,14)(11,15)(18,22)(19,23);(1,3,4,2)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24);(9,12)(10,15)(11,14)(13,16)(17,20)(18,23)(19,22)(21,24)Polycyclic presentation:
2^9|a^2=i,c^2=fgh,d^2=gh,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 | #72327388, #72327458, #72327517, … |
| maximal normal subgroup of | #72327388, #72327458, #72327517, … |
| maximal quotient of | #72329010, #72333086 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 442810);