Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454681 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,21)(12,26,30,17)(13,20)(14,18)(15,19)(16,23)(22,34,38,39)(24,32)(25,33)(27,31,37,40)(28,36,29,35);(1,3,4,7)(2,5,6,8)(9,11,15,24)(10,16,19,33)(12,27,30,37)(13,25,14,23)(17,34,26,39)(18,32,20,21)(22,28,38,29)(31,36,40,35);(3,7)(5,8)(9,12,13,29,15,30,14,28)(10,17,20,36,19,26,18,35)(11,22,25,37,24,38,23,27)(16,31,21,34,33,40,32,39);(10,18)(12,28)(13,14)(16,32)(19,20)(21,33)(22,37)(23,25)(27,38)(29,30)(31,40)(35,36)Polycyclic presentation:
2^9|a^2=eg,b^2=gh,c^2=ef,e^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,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381256);