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 | |
5056 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)(11,17,18,15)(12,16,20,21)(19,23,24,22)(25,26,30,27)(28,33,36,31)(29,34,37,32)(35,38,40,39);(2,6)(3,7)(10,14)(12,20)(15,17)(19,24)(25,26,30,27)(28,31,36,33)(29,34,37,32)(35,39,40,38);(1,3)(2,4)(5,7)(6,8)(11,18)(15,17)(19,24)(22,23)(26,27)(31,33)(32,34)(38,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72333433 |
| maximal normal subgroup of | #72333433 |
| maximal quotient of | #72333441, #72333468, #72334069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8659250);