Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453991 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
992 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)(3,5)(4,6)(7,8)(9,10,13,19,15,21,14,20)(11,22,24,17,26,33,25,16)(12,27,29,37,31,18,30,36)(23,35,28,32,39,40,38,34);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,23)(13,25)(14,24)(15,26)(17,21)(18,32)(19,33)(20,22)(27,34)(28,30)(29,38)(31,39)(35,36)(37,40);(2,6)(5,8)(9,11)(10,17)(12,28)(13,25)(14,24)(15,26)(16,21)(18,35)(19,22)(20,33)(23,29)(27,40)(30,39)(31,38)(32,37)(34,36);(9,12)(10,18)(11,23)(13,30)(14,29)(15,31)(16,32)(17,34)(19,37)(20,36)(21,27)(22,35)(24,38)(25,28)(26,39)(33,40)Polycyclic presentation:
2^9|a^2=e,b^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,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, 362952);