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 | |
4751 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,15,12)(11,13,16,18)(14,17,23,20)(19,21,22,24)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(1,3,5,8)(2,6,7,4)(9,11,15,16)(10,13,12,18)(14,22,23,19)(17,24,20,21)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,10,15,12)(11,18,16,13)(14,17,23,20)(19,24,22,21)(25,26,29,32)(27,30,35,33)(28,31,37,36)(34,38,40,39);(1,4)(2,3)(5,6)(7,8)(9,12,15,10)(11,13,16,18)(14,20,23,17)(19,21,22,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(9,13)(10,16)(11,12)(14,21)(15,18)(17,22)(19,20)(23,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(9,14)(10,17)(11,19)(12,20)(13,21)(15,23)(16,22)(18,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=gi,d^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72341056 |
| maximal normal subgroup of | #72341056 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10275812);