Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1600 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)(17,18,22,24,21,26,20,25)(19,27,30,23,28,31,29,32);(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,27)(25,31)(26,32);(2,4)(10,13)(14,16)(18,24)(19,28)(20,22)(23,31)(25,26)(27,32);(9,11,12,15)(10,14,13,16)(18,24)(20,22)(23,27)(25,26)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=gi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278182);