Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,11)(9,12)(13,14,19,24,20,25,18,33)(15,26,29,38,30,37,34,21)(16,22,17,23,31,40,32,39)(27,41,42,44,43,36,28,35);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,27)(17,28)(18,29)(19,34)(20,30)(22,35)(23,36)(24,37)(25,38)(26,33)(31,43)(32,42)(39,41)(40,44);(2,4)(5,7,9,8)(6,11,12,10)(13,16,20,31)(14,22,25,40)(15,27,30,43)(17,18,32,19)(21,35,38,44)(23,33,39,24)(26,41,37,36)(28,29,42,34);(2,4)(5,8,9,7)(6,11,12,10)(13,17)(14,23)(15,28)(16,18)(19,31)(20,32)(21,36)(22,33)(24,40)(25,39)(26,35)(27,29)(30,42)(34,43)(37,44)(38,41);(13,18,20,19)(14,24,25,33)(15,29,30,34)(16,17,31,32)(21,37,38,26)(22,39,40,23)(27,28,43,42)(35,41,44,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=gi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360368);