Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455572 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(6,8)(9,10)(11,22)(12,27,31,36)(13,20)(14,19)(15,21)(16,24)(17,25)(18,29,37,30)(23,32,39,34)(26,33)(28,35,38,40);(1,2,4,5)(3,6,7,8)(9,11,13,24,15,26,14,25)(10,16,19,22,21,17,20,33)(12,23,30,38,31,39,29,28)(18,32,27,35,37,34,36,40);(3,7)(6,8)(9,11)(10,17)(12,28)(13,25)(14,24)(15,26)(16,21)(18,35)(19,22)(20,33)(23,29)(27,32)(30,39)(31,38)(34,36)(37,40);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,23)(13,30)(14,29)(15,31)(16,32)(17,34)(19,27)(20,36)(21,37)(22,35)(24,38)(25,28)(26,39)(33,40)Polycyclic presentation:
2^9|a^2=fg,b^2=efg,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374998);