Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13657 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2177 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | C2×D32 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ #1311 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10,12,16,14,19,27,20)(11,21,25,32,23,15,31,33)(13,29,28,18,30,36,24,38)(17,37,35,22,39,40,34,26);(10,15)(11,22)(12,27)(13,28)(16,32)(17,38)(18,39)(19,21)(20,33)(23,26)(24,30)(25,37)(29,34)(31,40)(35,36);(1,3,4,8,5,2,6,7)(10,16,19,20)(11,23)(15,32,21,33)(17,34,39,35)(18,36,38,29)(22,26)(25,31)(37,40)Polycyclic presentation:
2^9|a^2=e,c^2=ehi,d^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fh,h^a=hi,d^b=dfi,e^b=ei,f^b=fh,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216311, #72223673, C2×#151085, … |
| maximal normal subgroup of | #72216311, #72223673, C2×#151085, … |
| maximal quotient of | C2×#151085, #72224652, #72230700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58281);