Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624888 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1333 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)(3,7)(4,6)(5,8)(10,16)(11,20)(12,15)(14,27)(17,30)(19,31)(21,26)(22,38)(23,37)(25,36)(28,33)(29,35)(32,40)(34,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,24)(13,16)(14,19)(15,18)(20,33)(21,31)(22,34)(23,28)(25,32)(26,29)(27,35)(30,37)(36,39)(38,40);(9,11,21,38,15,22,14,20,13,23,27,36,12,25,26,37)(10,17,31,40,18,34,19,33,16,28,35,39,24,32,29,30);(3,8)(5,7)(10,16)(17,28)(18,24)(19,29)(30,33)(31,35)(32,34)(39,40);(2,6)(5,7)(9,12,13,15)(10,18,16,24)(11,22,23,25)(14,21,26,27)(17,32,28,34)(19,35,29,31)(20,36,37,38)(30,39,33,40)Polycyclic presentation:
2^9|c^2=ghi,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72314254, #72315814, #72315829, … |
| maximal normal subgroup of | #72314254, #72315814, #72315829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532105);