Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476092 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(5,6,8,7)(10,16)(11,21)(13,14)(17,33)(18,35)(19,20)(23,37)(24,25)(26,32)(27,39)(28,29)(36,38);(1,2)(3,4)(5,6)(7,8)(9,10)(11,22)(12,26)(13,19)(14,16)(15,20)(17,24)(18,30)(21,33)(23,38)(25,31)(27,40)(28,35)(29,32)(34,37)(36,39);(2,4)(5,7,8,6)(9,11,13,21,15,25,14,24)(10,17,16,31,20,33,19,22)(12,27,28,39,30,37,29,23)(18,36,35,40,26,38,32,34);(6,7)(9,12,15,30)(10,18,20,26)(11,23,25,39)(13,29,14,28)(16,32,19,35)(17,34,33,40)(21,37,24,27)(22,36,31,38)Polycyclic presentation:
2^9|a^2=efi,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383726);