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 | |
4295 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)(3,7)(4,6)(5,8)(11,13)(12,20)(15,18)(16,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(1,2,4,6)(3,5,8,7)(9,10)(11,18)(12,16)(13,15)(14,17)(19,24)(20,23)(21,22)(26,31)(28,32)(34,39)(36,37);(1,2)(3,5)(4,6)(7,8)(9,11,14,13)(10,15,17,18)(12,21,20,19)(16,24,23,22)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(1,3)(2,5)(4,8)(6,7)(9,12,14,20)(10,16,17,23)(11,19,13,21)(15,22,18,24);(9,10)(11,15)(12,16)(13,18)(14,17)(19,22)(20,23)(21,24)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(9,13,14,11)(10,15,17,18)(12,21,20,19)(16,22,23,24)(25,29,30,38)(26,34,31,39)(27,35,33,40)(28,36,32,37)Polycyclic presentation:
2^9|b^2=hi,c^2=gi,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72219287 |
| maximal normal subgroup of | #72219283, #72219287 |
| maximal quotient of | #72219271 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9837991);