Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,13)(12,19)(14,15)(16,20)(17,23)(18,24)(21,22)(25,26)(27,29)(28,31)(30,36)(32,35)(33,38)(34,40)(37,39);(2,6)(3,7)(10,15)(11,17)(12,20)(22,24)(25,27,31,32)(26,29,28,35)(30,34,38,39)(33,37,36,40);(1,3)(2,4)(5,7)(6,8)(11,17)(13,23)(18,21)(22,24)(26,28)(27,32)(30,38)(37,40);(1,4)(2,3)(5,8)(6,7)(9,11)(10,13)(12,21)(14,17)(15,23)(16,24)(18,20)(19,22)(25,28,31,26)(27,29,32,35)(30,36,38,33)(34,37,39,40);(9,12)(10,16)(11,18)(13,22)(14,20)(15,19)(17,21)(23,24)(25,29)(26,32)(27,28)(30,37)(31,35)(33,39)(34,36)(38,40);(9,13)(10,11)(12,22)(14,23)(15,17)(16,18)(19,21)(20,24)(25,30)(26,33)(27,34)(28,36)(29,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
| Fitting subgroup of | #72340886 |
| maximal subgroup of | #72340883, #72340886, #72340899, … |
| maximal normal subgroup of | #72340883, #72340886, #72340899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841062);