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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,4,8,7)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24)(25,26)(27,30)(28,37)(29,33)(31,34)(32,36)(35,40)(38,39);(2,5)(3,8)(10,12)(15,19)(16,20)(22,24)(25,27)(26,30)(28,29)(31,35)(32,38)(33,37)(34,40)(36,39);(1,3,6,8)(2,7,5,4)(11,17)(13,21)(15,19)(16,20)(27,35)(28,36)(30,40)(32,37);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28)(26,32)(27,29)(30,38)(31,36)(33,40)(34,37)(35,39);(1,5,6,2)(3,4,8,7)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,29)(26,33)(27,36)(28,35)(30,32)(31,39)(34,38)(37,40);(1,4)(2,3)(5,8)(6,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,27)(28,38)(29,32)(31,40)(33,36)(34,35)(37,39)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219217, #72219281, #72219283 |
| maximal quotient of | #72219216 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8834186);