Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4390 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)(5,6,7,8)(9,10)(11,19)(12,16)(13,25)(14,18)(15,22)(17,28)(20,36)(21,34)(23,35)(24,33)(26,30)(27,32)(29,39)(31,38)(37,40);(9,11)(10,15)(12,20)(13,26)(14,21)(16,31)(17,34)(18,30)(19,25)(22,28)(23,39)(24,37)(27,38)(29,33)(32,40)(35,36);(2,4)(9,12,14,24)(10,16,18,33)(11,20,21,37)(13,27,28,23)(15,29,30,31)(17,35,25,32)(19,36,34,40)(22,39,26,38);(11,21)(15,30)(19,34)(20,37)(22,26)(29,31)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^b=ei,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72314588 |
| maximal normal subgroup of | #72314588 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264649);