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 | |
4542 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)(9,10,14,12)(11,15,18,16)(13,17,21,20)(19,22,24,23)(25,26)(27,28)(29,33)(30,32)(31,35)(34,36)(37,38)(39,40);(2,3)(6,7)(9,11)(10,15)(12,16)(13,19)(14,18)(17,22)(20,23)(21,24)(25,27)(26,31)(28,32)(29,37)(30,35)(33,36)(34,39)(38,40);(1,3,5,2)(4,7,8,6)(11,18)(15,16)(19,24)(22,23)(26,32)(28,31)(33,40)(36,38);(1,4,5,8)(2,6,3,7)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,28,30,31)(26,27,32,35)(29,38,39,36)(33,37,40,34);(9,12,14,10)(11,16,18,15)(13,17,21,20)(19,22,24,23)(25,29,30,39)(26,33,32,40)(27,34,35,37)(28,36,31,38);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,27)(26,31)(28,32)(29,37)(30,35)(33,36)(34,39)(38,40)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9110951);