Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #42882 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1827 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(6,9)(7,11)(13,14,19,25)(15,26,30,21)(16,31,33,22)(17,34,20,23)(18,35,36,24)(27,39,37,44)(28,40,38,42)(29,41,32,43);(5,6)(7,10)(8,9)(11,12)(13,15,19,30)(14,21,25,26)(16,18,33,36)(17,28,20,38)(22,35,31,24)(23,40,34,42)(27,29,37,32)(39,43,44,41);(2,4)(13,16,17,29,19,33,20,32)(14,22,34,43,25,31,23,41)(15,27,38,36,30,37,28,18)(21,39,40,24,26,44,42,35);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,23,25,34)(15,28,30,38)(16,32,33,29)(18,27,36,37)(21,40,26,42)(22,43,31,41)(24,39,35,44);(6,9)(10,12)(13,18,17,27,19,36,20,37)(14,24,34,44,25,35,23,39)(15,29,38,33,30,32,28,16)(21,41,40,22,26,43,42,31)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=f,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715154);