Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 | |
1376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18,22,26)(19,28,30,24)(20,31,29,32)(21,25,23,27);(10,13)(14,16)(17,19,22,30)(18,24,26,28)(20,21,29,23)(25,31,27,32);(1,2,3,4)(17,20,23,19,22,29,21,30)(18,24,25,32,26,28,27,31);(6,8)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,25,26,27)(19,29,30,20)(24,31,28,32)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=fi,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305498);