Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 | |
1190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(6,9)(10,11)(13,14,18,24,19,22,17,23)(15,25,27,20,29,36,28,37)(16,30,33,39,34,40,32,21)(26,42,43,38,44,35,31,41);(1,2)(3,4)(5,6,8,9)(7,11,12,10)(13,15)(14,20)(16,31)(17,27)(18,28)(19,29)(21,38)(22,37)(23,36)(24,25)(26,32)(30,42)(33,44)(34,43)(35,40)(39,41);(2,4)(13,16,19,34)(14,21,22,39)(15,26,29,44)(17,33,18,32)(20,35,37,42)(23,30,24,40)(25,41,36,38)(27,31,28,43);(5,7,8,12)(6,10,9,11)(14,22)(15,27)(17,18)(20,25)(21,39)(26,31)(28,29)(32,33)(35,41)(36,37)(38,42)(43,44)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337701);