Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1116 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)(19,22)(20,23)(21,24)(25,32)(26,30)(27,29)(28,31);(10,13)(14,15)(17,19,20,26)(18,22,23,30)(21,25,28,27)(24,32,31,29);(2,4)(9,11,12,16)(10,14,13,15)(18,23)(19,25)(21,28)(22,29)(26,27)(30,32);(5,7,8,6)(11,16)(14,15)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=e,d^2=efhi,g^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=de,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289141);