Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432838 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1573 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,38)(35,37)(39,40);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27)(26,30)(28,35)(29,34)(31,37)(32,38)(33,39)(36,40);(11,17)(14,20)(18,23)(21,24)(26,31)(28,36)(30,37)(32,33)(35,40)(38,39);(1,3,4,7)(2,6,5,8)(10,15)(14,20)(16,22)(21,24)(27,34)(30,38)(35,40)(37,39);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(26,32)(30,38)(31,33)(37,39)Polycyclic presentation:
2^9|d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,d^b=di,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308665, #72308719, #72308962, … |
| maximal normal subgroup of | #72308665, #72308719, #72308962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344220);