Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| 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 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,12,17)(11,20,21,15)(13,16,24,31)(14,27,25,19)(18,33,32,26)(22,34,37,28)(23,30,38,36)(29,39,35,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,21)(13,22)(14,23)(16,28)(17,20)(18,29)(19,30)(24,37)(25,38)(26,39)(27,36)(31,34)(32,35)(33,40);(1,4,5,2)(3,7,8,6)(10,16)(13,26)(14,25)(15,28)(17,31)(18,19)(20,34)(22,39)(23,38)(24,33)(27,32)(29,30)(35,36)(37,40);(3,8)(6,7)(11,21)(15,20)(22,37)(23,38)(28,34)(29,35)(30,36)(39,40);(10,17)(11,21)(16,31)(18,32)(19,27)(22,37)(23,38)(39,40)Polycyclic presentation:
2^9|a^2=f,c^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286697, #72320335, #72320356, … |
| maximal normal subgroup of | #72286697, #72320335, #72320356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531027);