Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394180 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(9,10,12,14)(11,15,16,13)(17,18,22,24)(19,25,27,32)(20,21,28,29)(23,30,31,26);(5,6)(7,8)(17,19,21,26,22,27,29,30)(18,23,20,25,24,31,28,32);(2,4)(6,8)(17,20,21,24,22,28,29,18)(19,25,26,31,27,32,30,23);(9,11,12,16)(10,13,14,15)(18,24)(20,28)(23,31)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72265008, #72265033, #72265115 |
| maximal normal subgroup of | #72265008, #72265033, #72265115 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322112);