Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453991 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,6,7,8)(9,10,14,12)(11,18,21,24)(13,22,16,17)(15,23,19,20);(2,5)(6,8)(9,11)(10,15)(12,19)(13,23)(14,21)(16,20)(17,24)(18,22);(1,3)(2,6)(4,7)(5,8)(9,12,13,22,14,10,16,17)(11,15,20,18,21,19,23,24);(3,7)(6,8)(9,12)(10,14)(11,19)(13,17)(15,21)(16,22)(18,20)(23,24)Polycyclic presentation:
2^9|a^2=ef,c^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72227842, #72227844, #72227849 |
| maximal normal subgroup of | #72227842, #72227844, #72227849 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364791);