Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200424 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(5,8)(6,7)(9,10,13,18,14,17,19,32)(11,15,22,29,21,30,26,33)(12,23,25,34,24,16,27,31)(20,28,36,39,35,38,37,40);(1,3,4,2,6,8,5,7)(9,11,19,22,14,21,13,26)(10,15,18,33,17,30,32,29)(12,20,25,37,24,35,27,36)(16,28,34,39,23,38,31,40);(9,12,14,24)(10,16,17,23)(11,20,21,35)(13,27,19,25)(15,28,30,38)(18,34,32,31)(22,37,26,36)(29,40,33,39);(1,4,6,5)(2,3,7,8)(10,17)(11,21)(16,23)(18,32)(20,35)(22,26)(31,34)(36,37)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=i,d^2=hi,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107792);