Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6)(7,8)(10,11)(14,15)(17,18,21,26,20,25,22,27)(19,28,30,32,29,23,31,24);(1,2)(3,4)(9,10,13,11)(12,14,16,15)(17,19,21,30,20,29,22,31)(18,23,26,24,25,28,27,32);(2,4)(9,11,13,10)(12,14,16,15)(17,19,20,29)(18,24,25,32)(21,31,22,30)(23,27,28,26);(6,8)(9,12,13,16)(10,14,11,15)(18,25)(19,29)(26,27)(30,31)Polycyclic presentation:
2^9|a^2=efi,b^2=f,c^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364627);