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 | |
4488 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)(3,4)(5,6)(7,8)(10,15)(13,23)(16,22)(17,24)(28,35)(32,37)(34,39)(38,40);(2,6)(3,8)(11,19)(12,21)(16,22)(17,24)(26,30)(27,33)(32,37)(34,39);(1,3,5,8)(2,7,6,4)(9,10,14,15)(11,16,19,22)(12,17,21,24)(13,18,23,20)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,3,5,8)(2,7,6,4)(9,11)(10,16)(12,20)(13,17)(14,19)(15,22)(18,21)(23,24);(1,2)(3,7)(4,8)(5,6)(9,12,14,21)(10,17,15,24)(11,20,19,18)(13,22,23,16)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,4,5,7)(2,8,6,3)(9,13)(10,18)(11,17)(12,22)(14,23)(15,20)(16,21)(19,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|c^2=ghi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219279, #72219281, #72219283 |
| maximal quotient of | #72219277 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000431);