Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
17065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,20)(12,17)(13,18)(14,27)(15,19)(16,22)(21,34)(23,30)(24,32)(25,33)(26,37)(28,31)(29,36)(35,40)(38,39);(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,29)(18,30)(19,31)(20,27)(24,35)(25,36)(26,38)(32,39)(33,40)(34,37);(2,4)(9,12)(10,17)(11,21)(13,25)(14,26)(15,24)(16,29)(18,33)(19,32)(20,34)(22,36)(23,35)(27,37)(28,38)(30,40)(31,39);(6,8)(12,24)(17,32)(21,35)(25,26)(29,39)(33,37)(34,40)(36,38);(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,25,24,26)(16,30,31,20)(17,33,32,37)(21,36,35,38)(29,40,39,34)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,f^b=fi,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286571, #72315582, #72315585, … |
| maximal normal subgroup of | #72286571, #72315582, #72315585, … |
| maximal quotient of | #72286750 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536037);