Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60797 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1903 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)(9,10,14,19,15,12,24,18)(11,16,23,31,21,20,35,30)(13,17,28,34,27,25,37,33)(22,29,36,40,26,32,38,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,20)(13,26)(14,23)(15,21)(17,32)(18,30)(19,31)(22,27)(24,35)(25,29)(28,38)(33,40)(34,39)(36,37);(3,8)(5,7)(9,12)(10,15)(11,20)(13,25)(14,18)(16,21)(17,27)(19,24)(22,32)(23,30)(26,29)(28,33)(31,35)(34,37)(36,39)(38,40);(2,6)(5,7)(11,21)(13,27)(16,20)(17,25)(23,35)(28,37)(30,31)(33,34);(9,13,14,28,15,27,24,37)(10,17,19,34,12,25,18,33)(11,22,23,36,21,26,35,38)(16,29,31,40,20,32,30,39)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di,e^b=ei,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6285057);