Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #28483 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,11)(7,12,8,10)(13,14,20,26)(15,21,29,38)(16,22,27,36)(17,23,19,25)(18,32,35,24)(28,43,44,37)(30,41,33,39)(31,42,34,40);(5,6)(7,10)(8,12)(9,11)(13,15,20,29)(14,21,26,38)(16,17,27,19)(18,33,35,30)(22,23,36,25)(24,41,32,39)(28,31,44,34)(37,40,43,42);(1,2,3,4)(13,16,19,15,20,27,17,29)(14,22,25,21,26,36,23,38)(18,33,34,44,35,30,31,28)(24,41,42,43,32,39,40,37);(5,7,9,8)(6,10,11,12)(13,17,20,19)(14,23,26,25)(15,27,29,16)(18,31,35,34)(21,36,38,22)(24,40,32,42)(28,33,44,30)(37,41,43,39);(5,8,9,7)(6,10,11,12)(13,18,17,31,20,35,19,34)(14,24,23,40,26,32,25,42)(15,28,16,30,29,44,27,33)(21,37,22,39,38,43,36,41)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gh,d^2=fi,e^2=fhi,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=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724054);