Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×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 | |
1798 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)(9,10,12,13)(11,14,15,16)(17,18)(19,27)(20,28)(21,24)(22,26)(23,30)(25,31)(29,32);(1,2,3,4)(5,6)(7,8)(10,13)(11,15)(17,19,21,28,22,29,31,23)(18,20,25,27,26,30,24,32);(6,8)(10,13)(14,16)(17,20,22,30)(18,23,26,28)(19,24,29,25)(21,27,31,32);(2,4)(9,11,12,15)(10,14,13,16)(18,24)(19,28)(21,31)(23,29)(25,26)(27,32)Polycyclic presentation:
2^9|a^2=fi,b^2=efi,c^2=i,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=dei,e^d=ei,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337250);