Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,3,4)(6,8)(9,10,12,14)(11,15,16,13)(17,18,20,25,21,24,30,23)(19,26,28,31,29,32,27,22);(2,4)(9,11,12,16)(10,13,14,15)(17,19)(18,22)(20,27)(21,29)(23,26)(24,31)(25,32)(28,30);(2,4)(18,23)(19,27)(20,30)(24,25)(26,32)(28,29);(5,6,7,8)(18,24)(22,31)(23,25)(26,32)Polycyclic presentation:
2^9|a^2=e,b^2=efi,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237292, #72238785, #72239631, … |
| maximal normal subgroup of | #72237292, #72238785, #72239631, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 320001);