Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395409 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,22,14,16,13,17)(11,18,21,23,20,24,19,15);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,23)(17,24)(18,22);(10,16)(11,19)(12,13)(15,18)(20,21)(23,24);(2,6)(5,8)Polycyclic presentation:
2^9|a^2=ef,b^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251849, #72251857, #72251864, … |
| maximal normal subgroup of | #72251849, #72251857, #72251864, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312983);