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 | |
4644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(25,26)(27,33)(28,29)(30,34)(31,37)(32,38)(35,40)(36,39);(3,7)(6,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27)(26,31)(28,35)(29,36)(30,37)(32,40)(33,34)(38,39);(1,2,5,4)(3,6,7,8)(10,11)(13,21)(15,17)(19,24)(28,38)(29,32)(35,39)(36,40);(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,28)(26,32)(27,35)(29,34)(30,38)(31,40)(33,36)(37,39);(1,3,5,7)(2,6,4,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,27)(26,33)(28,39)(29,36)(30,37)(31,34)(32,40)(35,38);(1,4,5,2)(3,8,7,6)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,32)(26,28,34,38)(27,36,37,40)(31,35,33,39)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=i,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219279, #72219285 |
| maximal quotient of | #72219270 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8830889);