Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184831 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1092 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)(3,7)(4,6)(5,8)(17,18,23,27,22,26,21,25)(19,24,28,31,20,29,30,32);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,22)(21,30)(23,28)(25,32)(26,29)(27,31);(10,13)(14,16)(17,20,22,19)(18,24,26,29)(21,30,23,28)(25,31,27,32);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,25,26,27)(19,28,20,30)(24,31,29,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=i,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270515);