Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198510 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60958 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,7,6,3,4,8)(9,10,14,18,15,19,13,29)(11,20,24,31,22,35,23,16)(12,25,27,17,28,32,26,33)(21,30,38,34,36,40,37,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16,19,31)(12,21)(13,23)(14,24)(15,22)(17,30,33,40)(18,20,29,35)(25,39,32,34)(26,37)(27,38)(28,36);(1,2)(3,6)(4,7)(5,8)(9,12)(10,17)(11,21)(13,26)(14,27)(15,28)(16,30)(18,32)(19,33)(20,34)(22,36)(23,37)(24,38)(25,29)(31,40)(35,39);(1,4,6,5)(2,7,3,8)(11,22)(16,31)(20,35)(21,36)(23,24)(30,40)(34,39)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=i,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319604 |
| maximal normal subgroup of | #72319604 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105938);