Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2754 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)(7,12)(8,10)(9,11)(13,14,20,24,19,26,18,25)(15,27,30,21,32,40,31,37)(16,22,17,23,34,39,33,38)(28,36,43,35,42,44,29,41);(1,2)(3,4)(13,15)(14,21)(16,28)(17,29)(18,30)(19,32)(20,31)(22,35)(23,36)(24,27)(25,40)(26,37)(33,43)(34,42)(38,44)(39,41);(2,4)(5,7,9,8)(6,10,11,12)(13,16,19,34)(14,22,26,39)(15,28,32,42)(17,18,33,20)(21,35,37,41)(23,25,38,24)(27,36,40,44)(29,30,43,31);(2,4)(5,8,9,7)(6,10,11,12)(13,17)(14,23)(15,29)(16,18)(19,33)(20,34)(21,36)(22,25)(24,39)(26,38)(27,41)(28,30)(31,42)(32,43)(35,40)(37,44);(13,18,19,20)(14,24,26,25)(15,30,32,31)(16,17,34,33)(21,27,37,40)(22,38,39,23)(28,29,42,43)(35,44,41,36)Polycyclic presentation:
2^9|a^2=fi,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, 6359214);