Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2948 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)(13,14,17,22,18,21,30,35)(15,23,25,32,26,36,29,19)(16,20,28,34,27,33,40,43)(24,37,39,42,38,44,41,31);(5,6,8,9)(7,10,11,12)(13,15,18,26)(14,19,21,32)(16,24,27,38)(17,29,30,25)(20,31,33,42)(22,36,35,23)(28,41,40,39)(34,44,43,37);(2,4)(6,9)(7,11)(13,16)(14,20)(15,24)(17,28)(18,27)(19,31)(21,33)(22,34)(23,37)(25,39)(26,38)(29,41)(30,40)(32,42)(35,43)(36,44);(5,7)(6,10)(8,11)(9,12)(14,21)(16,27)(19,32)(22,35)(23,36)(24,38)(28,40)(39,41);(6,9)(7,11)(16,27)(20,33)(24,38)(28,40)(31,42)(34,43)(37,44)(39,41)Polycyclic presentation:
2^9|a^2=f,b^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503322);