Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
545 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,13)(15,16);(1,3,4,8)(2,5,6,7)(17,18,19,24,27,31,32,29,21,23,28,30,20,25,26,22);(10,13)(14,16)(18,22)(19,26)(20,27)(23,29)(24,25)(28,32)(30,31);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,23)(22,29)(24,30)(25,31)Polycyclic presentation:
2^9|a^2=e,b^2=eg,d^2=f,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72343635, #72343815, #72343846 |
| maximal normal subgroup of | #72343635, #72343815, #72343846 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 414332);