Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,24)(21,23)(22,29)(25,30)(27,32)(28,31);(1,2,3,4)(5,6)(7,8)(17,19,20,29,21,27,30,28)(18,22,25,26,23,31,24,32);(6,8)(10,13)(14,16)(18,23)(19,27)(24,25)(28,29);(2,4)(9,11,12,15)(10,14,13,16)(18,24)(19,28)(20,30)(22,31)(23,25)(27,29)Polycyclic presentation:
2^9|b^2=e,d^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^d=ei,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72311829, #72311837, #72311868, … |
| maximal normal subgroup of | #72311829, #72311837, #72311868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334749);