Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187766 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,6,3,8)(9,10)(11,21,24,34)(12,25,28,17)(13,18)(14,19)(15,20)(16,23,31,29)(22,36)(26,33,27,32)(30,37)(35,38)(39,40);(2,6)(3,4)(9,11)(10,16)(12,22)(13,23)(14,29)(15,24)(17,30)(18,21)(19,34)(20,31)(25,39)(26,37)(27,40)(28,38)(32,35)(33,36);(9,12)(10,17)(11,22)(13,26)(14,27)(15,28)(16,30)(18,32)(19,33)(20,25)(21,35)(23,37)(24,38)(29,40)(31,39)(34,36);(1,3,5,4)(2,7,6,8)(9,13,15,14)(10,18,20,19)(11,23,24,29)(12,26,28,27)(16,21,31,34)(17,32,25,33)(22,37,38,40)(30,35,39,36)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,d^b=dh,e^b=ei,g^b=ghi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306312, #72306314, #72306317, … |
| maximal normal subgroup of | #72306312, #72306314, #72306317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95039);