Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2619 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16662 |
| prime radical | #23434 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1037 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,15)(4,21)(5,11)(6,13)(7,12)(8,14)(9,18)(10,23)(16,27)(17,28)(19,29)(20,32)(22,25)(24,30)(26,31);(1,3,6,18,8,20,7,19)(2,9,12,15,14,29,13,32)(4,22,23,11,24,31,28,16)(5,17,25,30,27,10,26,21);(1,4,7,23,8,24,6,28)(2,10,13,21,14,17,12,30)(3,16,18,31,20,11,19,22)(5,15,26,29,27,32,25,9);(1,5)(2,11,14,16)(3,17)(4,9,24,29)(6,25)(7,26)(8,27)(10,20)(12,31,13,22)(15,28,32,23)(18,30)(19,21)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,d^c=de,e^c=ei,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72339028, #72339034, #72339041 |
| maximal normal subgroup of | #72339028, #72339034, #72339041 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376962);