Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(5,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,27)(20,23)(21,26)(22,25)(24,30)(28,31)(29,32);(5,7,8,6)(10,13)(14,15)(18,23)(19,28)(21,22)(24,27)(29,30)(31,32);(2,4)(17,19,20,29)(18,24,23,31)(21,28,22,30)(25,32,26,27);(9,11,12,16)(10,14,13,15)(18,23)(19,29)(25,26)(28,30)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272071);