Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #42652 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1857 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)(13,14,20,26)(15,21,28,35)(16,22,32,39)(17,23,19,25)(18,24,33,40)(27,34,41,43)(29,36,42,44)(30,37,31,38);(5,6)(7,10)(8,9)(11,12)(13,15,20,28)(14,21,26,35)(16,29,32,42)(17,18,19,33)(22,36,39,44)(23,24,25,40)(27,30,41,31)(34,37,43,38);(2,4)(13,16,19,31,20,32,17,30)(14,22,25,38,26,39,23,37)(15,27,18,29,28,41,33,42)(21,34,24,36,35,43,40,44);(5,7,8,11)(6,10,9,12)(13,17,20,19)(14,23,26,25)(15,18,28,33)(16,30,32,31)(21,24,35,40)(22,37,39,38)(27,29,41,42)(34,36,43,44);(6,9)(10,12)(13,18,19,28,20,33,17,15)(14,24,25,35,26,40,23,21)(16,29,31,41,32,42,30,27)(22,36,38,43,39,44,37,34)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=fi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715357);