Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
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 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2652 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)(5,6,7,8)(17,18,22,26)(19,27,30,31)(20,25,21,24)(23,29,32,28);(2,4)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,26,32)(20,29,21,28)(24,31,25,27);(6,8)(10,13)(14,16)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29);(9,11,12,15)(10,14,13,16)(18,25)(20,21)(23,27)(24,26)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=efi,b^2=i,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,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=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72264361, #72264385, #72264497, … |
| maximal normal subgroup of | #72264361, #72264385, #72264497, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311447);