Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator | C22 | C23×C42 |
| socle | C23 | C24×C4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(9,10,13,18,14,17,24,30)(11,15,21,26,20,27,33,37)(12,16,23,29,22,28,34,38)(19,25,32,36,31,35,39,40);(2,4)(5,6,8,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,25)(17,27)(18,26)(22,31)(23,32)(24,33)(28,35)(29,36)(30,37)(34,39)(38,40);(5,7,8,6)(12,22)(16,28)(19,31)(23,34)(25,35)(29,38)(32,39)(36,40);(9,12,13,23,14,22,24,34)(10,16,18,29,17,28,30,38)(11,19,21,32,20,31,33,39)(15,25,26,36,27,35,37,40);(10,17)(11,20)(16,28)(18,30)(19,31)(21,33)(29,38)(32,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=g,g^2=i,b^a=bf,e^a=ei,e^b=ei,d^c=di176 conjugacy classes — view the full character table.
| maximal subgroup of | #72343474, #72343481, #72343485, … |
| maximal normal subgroup of | #72343474, #72343481, #72343485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277231);