Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395151 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,29,30,26)(20,32,31,25)(22,28,23,27);(5,6)(7,8)(9,11,12,16)(10,13,14,15)(18,24)(25,32)(26,29)(27,28);(1,2,3,4)(6,8)(17,19,22,31,21,30,23,20)(18,25,27,29,24,32,28,26);(17,20,23,30,21,31,22,19)(18,26,28,32,24,29,27,25)Polycyclic presentation:
2^9|a^2=e,b^2=ei,c^2=f,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg104 conjugacy classes — view the full character table.
| maximal subgroup of | #72265215, #72265225, #72265262 |
| maximal normal subgroup of | #72265215, #72265225, #72265262 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 320448);