Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417647 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,8)(5,7)(9,10,12,14)(11,13,15,16)(18,23)(19,22)(20,27)(24,26)(28,31)(29,30);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,26)(19,24,22,23)(20,28,30,31)(25,27,32,29);(2,7)(6,8)(11,15)(13,16)(17,19,21,22)(18,24,26,23)(20,27,30,29)(25,28,32,31);(1,4,5,3)(2,8,7,6)(17,20)(18,25)(19,27)(21,30)(22,29)(23,31)(24,28)(26,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343639);