Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C4×D16 |
| prime radical | C2×D32 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ C82 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
158 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)(11,15,22,28)(13,26,24,40)(14,27,25,19)(17,33,31,20)(18,35,32,21)(23,36,37,30)(29,39,34,38);(10,15)(11,20)(13,21)(14,25)(16,28)(17,18)(19,36)(22,33)(23,39)(24,35)(26,29)(27,30)(31,32)(34,40)(37,38);(1,3,4,8,5,2,6,7)(10,16)(15,28)(17,31)(18,32)(19,27)(26,40)(29,34)(30,36)Polycyclic presentation:
2^9|a^2=e,c^2=ei,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,h^a=hi,d^b=df,f^b=fh,h^b=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#2263, C2×#151080, #72230700, … |
| maximal normal subgroup of | C8×#2263, C2×#151080, #72230700, … |
| maximal quotient of | C2×#151080, #72230701, #72230739 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58276);