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 | |
4557 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)(11,13)(12,20)(14,15)(16,21)(17,18)(19,24)(22,23)(25,26,31,32)(27,29,28,33)(30,34,38,39)(35,37,36,40);(2,3)(6,7)(9,11)(10,13)(12,19)(14,17)(15,18)(16,22)(20,24)(21,23)(26,32)(27,28)(34,39)(35,36);(1,3,5,2)(4,7,8,6)(10,15)(11,17)(12,21)(22,24)(25,27,31,28)(26,33,32,29)(30,35,38,36)(34,40,39,37);(1,4)(2,6)(3,7)(5,8)(9,10)(11,18)(12,16)(13,17)(14,15)(19,24)(20,21)(22,23)(25,28,31,27)(26,29,32,33)(30,36,38,35)(34,37,39,40);(9,12,14,21)(10,16,15,20)(11,19,17,23)(13,22,18,24)(25,29)(26,27)(28,32)(30,37)(31,33)(34,35)(36,39)(38,40);(9,13,14,18)(10,17,15,11)(12,22,21,24)(16,23,20,19)(25,30,31,38)(26,34,32,39)(27,35,28,36)(29,37,33,40)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8038415);