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 | |
4930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,14)(12,21)(13,17)(15,18)(16,20)(19,24)(22,23)(25,26,30,31)(27,33,35,28)(29,32,38,39)(34,37,40,36);(2,3)(6,7)(9,11)(10,14)(12,13)(15,19)(16,22)(17,21)(18,24)(20,23)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(1,3,5,2)(4,7,8,6)(11,19)(12,20)(14,24)(16,21)(26,31)(28,33)(32,39)(36,37);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,13)(14,22)(15,20)(17,24)(18,21)(19,23)(25,28)(26,27)(29,37)(30,33)(31,35)(32,40)(34,39)(36,38);(9,13)(10,17)(11,20)(12,19)(14,16)(15,23)(18,22)(21,24)(25,29,30,38)(26,32,31,39)(27,34,35,40)(28,36,33,37);(9,14)(10,11)(12,22)(13,16)(15,24)(17,20)(18,19)(21,23)(25,26)(27,28)(29,32)(30,31)(33,35)(34,36)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=g,c^2=gh,e^2=h,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 | #72219217, #72219279 |
| maximal quotient of | #72219214 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8834896);