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 | |
4614 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)(11,18)(12,13)(14,15)(16,21)(17,22)(19,20)(23,24)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,40)(38,39);(2,3)(6,7)(10,15)(12,22)(16,18)(19,24)(27,36)(28,38)(31,32)(33,39);(1,3,5,2)(4,7,8,6)(25,27)(26,31)(28,37)(29,33)(30,36)(32,35)(34,38)(39,40);(1,4)(2,6)(3,7)(5,8)(9,11,14,21)(10,16,15,18)(12,19,22,24)(13,23,17,20)(25,27)(26,32)(28,37)(29,39)(30,36)(31,35)(33,40)(34,38);(9,12)(10,17)(11,19)(13,15)(14,22)(16,20)(18,23)(21,24)(25,28)(26,33)(27,37)(29,31)(30,38)(32,40)(34,36)(35,39);(9,13,14,17)(10,12,15,22)(11,20,21,23)(16,24,18,19)(25,29,30,40)(26,34,35,37)(27,33,36,39)(28,31,38,32)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8412261);