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 | #36967 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(12,19)(16,21)(17,23)(22,24)(25,26,31,32)(27,35,39,33)(28,30,29,34)(36,40,37,38);(2,3)(6,7)(10,14)(15,20)(16,21)(22,24)(26,32)(28,29)(33,35)(36,37);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,27,31,39)(26,33,32,35)(28,36,29,37)(30,38,34,40);(1,4)(2,6)(3,7)(5,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,28,31,29)(26,34,32,30)(27,36,39,37)(33,40,35,38);(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,29,31,28)(26,30,32,34)(27,37,39,36)(33,38,35,40);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,30)(26,28)(27,38)(29,32)(31,34)(33,36)(35,37)(39,40)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219287 |
| maximal normal subgroup of | #72219279, #72219287 |
| maximal quotient of | #72219274 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8016456);