Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×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 | C23×C22⋊C4 |
| prime radical | #62815 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10)(11,15)(12,23,25,33)(13,18)(14,19)(16,24,31,26)(17,21)(20,35,36,40)(22,29)(27,34)(28,37,30,39)(32,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,28)(17,19)(18,29)(23,35)(24,37)(25,36)(26,39)(27,38)(30,31)(32,34)(33,40);(2,5)(6,8)(9,12)(10,16)(11,20)(13,26)(14,25)(15,28)(17,30)(18,33)(19,31)(21,36)(22,39)(23,34)(24,27)(29,40)(32,35)(37,38);(1,3,4,7)(2,6,5,8)(9,11)(10,17)(12,20)(13,22)(14,21)(15,19)(16,30)(18,32)(23,40)(24,37)(25,36)(26,39)(27,38)(28,31)(29,34)(33,35);(11,21)(15,17)(20,36)(22,38)(28,30)(29,32)(35,40)(37,39)Polycyclic presentation:
2^9|a^2=h,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=cf,e^b=ei,g^c=gi,h^c=hi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72303255, #72303270, #72303398 |
| maximal normal subgroup of | #72303255, #72303270, #72303398 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281624);