Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1800 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)(3,7,8,5)(4,6)(9,10)(11,19,22,15)(12,23,24,38)(13,18)(14,17)(16,25,29,39)(20,33)(21,32,36,28)(26,30)(27,35)(31,40)(34,37);(9,11)(10,15)(12,20)(13,21)(14,22)(16,27)(17,19)(18,28)(23,37)(24,34)(25,35)(26,36)(29,31)(30,32)(33,38)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16,17,29)(11,20)(13,25)(14,24)(15,27,19,31)(18,23,30,38)(21,35)(22,34)(26,39)(28,37,32,33)(36,40);(12,24)(16,29)(20,34)(23,38)(25,39)(27,31)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=g,c^2=e,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,d^c=dh,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72255852, #72255859, #72255866, … |
| maximal normal subgroup of | #72255852, #72255859, #72255866, … |
| maximal quotient of | #72262592 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97251);