Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420064 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2420 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,22)(12,28,32,18)(13,20)(14,19)(15,21)(16,25)(17,31,38,30)(23,33,24,39)(26,36)(27,35)(29,34,40,37);(2,5)(3,7)(9,11,14,26,15,27,13,25)(10,16,20,35,21,36,19,22)(12,29,31,24,32,40,30,23)(17,37,18,39,38,34,28,33);(9,12)(10,17)(11,23)(13,31)(14,30)(15,32)(16,33)(18,19)(20,28)(21,38)(22,37)(24,27)(25,29)(26,40)(34,35)(36,39);(1,3,4,7)(2,6,5,8)(9,12)(10,18,21,28)(11,24)(13,31)(14,30)(15,32)(16,34,36,37)(17,19,38,20)(22,33,35,39)(23,27)(25,40)(26,29)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 360181);