Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(6,7)(9,10,12,13)(11,14,15,16)(17,18)(19,26)(20,22)(21,24)(23,27)(25,29)(28,30)(31,32);(1,2,3,4)(5,6,8,7)(10,13)(14,16)(18,22)(19,27)(24,25)(28,32);(2,4)(5,6)(7,8)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32);(5,7,8,6)(11,15)(14,16)(18,24)(19,27)(21,29)(22,25)(23,30)(26,31)Polycyclic presentation:
2^9|a^2=ef,b^2=gi,c^2=ef,d^2=fi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374164);