Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393904 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(6,7)(9,10)(11,21)(12,26)(13,19)(14,18)(15,20)(16,24)(17,29)(22,38)(23,33)(25,32)(27,37)(28,36)(30,35)(31,39)(34,40);(5,6,8,7)(9,11)(10,16)(12,27)(13,25)(14,24)(15,23)(17,34)(18,33)(19,21)(20,32)(22,29)(26,38)(28,39)(30,40)(31,35)(36,37);(9,12,15,30)(10,17,20,36)(11,22,23,39)(13,29,14,28)(16,31,32,38)(18,26,19,35)(21,37,33,34)(24,27,25,40);(2,4)(5,7,8,6)(11,23)(16,32)(21,33)(22,39)(24,25)(27,40)(31,38)(34,37)Polycyclic presentation:
2^9|b^2=ef,c^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307869);