Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394046 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(9,10)(11,22)(12,17)(13,20)(14,19)(15,21)(16,24)(18,26)(23,31)(25,35)(27,34)(28,32)(29,33)(30,38)(36,39)(37,40);(1,2,4,6)(3,7,8,5)(9,11)(10,16)(12,23)(13,28)(14,24)(15,25)(17,30)(18,31)(19,35)(20,22)(21,32)(26,40)(27,39)(29,38)(33,36)(34,37);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,23)(13,29)(14,26)(15,27)(16,30)(18,19)(20,33)(21,34)(22,36)(24,40)(25,39)(28,38)(31,35)(32,37);(9,12)(10,18)(11,23)(13,26)(14,29)(15,27)(16,31)(17,19)(20,34)(21,33)(22,37)(24,38)(25,39)(28,40)(30,35)(32,36)Polycyclic presentation:
2^9|b^2=g,c^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282010 |
| maximal normal subgroup of | #72282010 |
| maximal quotient of | #72282194 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310566);