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 | |
5001 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,34)(28,32)(29,37)(30,31)(33,38)(35,36)(39,40);(2,3)(6,7)(11,17)(15,21)(18,23)(22,24)(26,31)(28,34)(33,37)(36,40);(1,3,5,2)(4,7,8,6)(10,14)(15,21)(16,19)(22,24)(26,31)(27,32)(29,38)(36,40);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,40)(38,39);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28)(26,32)(27,31)(29,36)(30,34)(33,39)(35,37)(38,40);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=i,c^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72333433 |
| maximal normal subgroup of | #72333433 |
| maximal quotient of | #72333448, #72333462, #72333482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9350373);