Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×#509 |
| 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 |
| prime radical | #33582 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,7)(9,10,15,20)(11,16,14,18)(12,23,25,39)(13,26,29,35)(17,32,34,24)(19,22,37,28)(21,38,30,36)(27,31,40,33);(1,2,4,5)(3,7,8,6)(9,11,15,14)(10,16,20,18)(12,21,25,30)(13,22,29,28)(17,31,34,33)(19,26,37,35)(23,38,39,36)(24,27,32,40);(9,12)(10,17)(11,21)(13,27)(14,30)(15,25)(16,31)(18,33)(19,36)(20,34)(22,32)(23,26)(24,28)(29,40)(35,39)(37,38);(1,3,4,8)(2,6,5,7);(9,11,15,14)(10,18,20,16)(12,21,25,30)(13,22,29,28)(17,33,34,31)(19,35,37,26)(23,36,39,38)(24,27,32,40)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72309713, #72309719, #72309732, … |
| maximal normal subgroup of | #72309713, #72309719, #72309732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531079);