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 | |
1432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,27)(20,24)(21,26)(22,25)(23,30)(28,32)(29,31);(5,7,8,6)(10,13)(14,15)(17,19,21,30)(18,23,26,27)(20,29,22,28)(24,32,25,31);(1,2,3,4)(18,24)(19,28)(20,22)(23,27)(25,26)(29,30);(9,11,12,16)(10,14,13,15)(17,20,21,22)(18,25,26,24)(19,29,30,28)(23,31,27,32)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=fg,d^2=e,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, 276197);