Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453948 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1233 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)(18,34)(19,20)(22,30)(23,26)(24,25)(27,28)(31,36)(32,40)(37,38);(1,2)(3,4)(5,7,8,6)(9,10)(11,22)(12,18)(13,19)(14,16)(15,20)(17,21)(23,35)(24,33)(25,30)(26,32)(27,31)(28,34)(29,36)(37,40)(38,39);(2,4)(6,7)(9,11,13,24,15,25,14,21)(10,17,16,30,20,33,19,22)(12,26,27,38,29,37,28,23)(18,35,34,40,36,39,31,32);(9,12,15,29)(10,18,20,36)(11,23,25,38)(13,28,14,27)(16,31,19,34)(17,32,33,40)(21,26,24,37)(22,35,30,39)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,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,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364096);