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×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,13,14)(11,15,17,18)(12,16,20,21)(19,22,23,24)(25,26,31,36)(27,34,29,32)(28,38,39,33)(30,35,37,40);(2,6)(3,7)(10,14)(11,17)(16,21)(19,23)(25,27,31,29)(26,32,36,34)(28,30,39,37)(33,35,38,40);(1,3)(2,4)(5,7)(6,8)(11,17)(12,20)(15,18)(16,21)(25,28,31,39)(26,33,36,38)(27,37,29,30)(32,40,34,35);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,29,31,27)(26,34,36,32)(28,30,39,37)(33,35,38,40);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,30)(26,35)(27,28)(29,39)(31,37)(32,33)(34,38)(36,40);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^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 | #72340914, #72340915, #72341116 |
| maximal subgroup of | #72340911, #72340914, #72340944, … |
| maximal normal subgroup of | #72340911, #72340914, #72340944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840387);