Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18,22,23,21,26,20,25)(19,27,30,32,29,24,28,31);(1,2)(3,4)(5,7,8,6)(18,23)(20,22)(24,31)(25,26)(27,32)(28,30);(2,4)(10,13)(14,16)(17,19,21,29)(18,24,26,27)(20,28,22,30)(23,31,25,32);(9,11,12,15)(10,14,13,16)(18,25)(20,22)(23,26)(24,32)(27,31)(28,30)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=i,d^2=fgi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=ei,h^b=hi,e^d=ei,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279603);