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 | |
4376 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)(3,5)(4,7)(6,8)(11,17)(15,22)(18,20)(23,24)(25,26,31,35)(27,36,40,32)(28,33,38,30)(29,34,37,39);(1,2,6,8)(3,5,7,4)(9,10)(11,15)(12,19)(13,14)(16,21)(17,22)(18,24)(20,23)(25,27,31,40)(26,32,35,36)(28,37,38,29)(30,39,33,34);(1,3,6,7)(2,4,8,5)(10,14)(11,17)(16,19)(18,20)(25,28,31,38)(26,33,35,30)(27,37,40,29)(32,34,36,39);(1,4,6,5)(2,3,8,7)(9,11)(10,15)(12,20)(13,17)(14,22)(16,24)(18,21)(19,23)(25,28)(26,33)(27,37)(29,40)(30,35)(31,38)(32,34)(36,39);(1,2)(3,4)(5,7)(6,8)(9,12,13,21)(10,16,14,19)(11,18,17,20)(15,23,22,24)(25,29,31,37)(26,34,35,39)(27,38,40,28)(30,32,33,36);(1,5,6,4)(2,7,8,3)(9,10)(11,15)(12,16)(13,14)(17,22)(18,23)(19,21)(20,24)(25,30,31,33)(26,28,35,38)(27,39,40,34)(29,32,37,36)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219283, #72219285 |
| maximal quotient of | #72219275 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8838233);