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 | |
4465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,18)(12,16)(13,15)(14,17)(19,24)(20,22)(21,23)(25,26,29,34)(27,33,37,30)(28,32,38,31)(35,39,36,40);(1,2)(3,7)(4,8)(5,6)(9,11,14,13)(10,15,17,18)(12,19,22,21)(16,23,20,24)(25,27)(26,30)(28,36)(29,37)(31,40)(32,39)(33,34)(35,38);(11,13)(15,18)(19,21)(23,24)(25,28,29,38)(26,31,34,32)(27,35,37,36)(30,39,33,40);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,16,17,20)(11,19,13,21)(15,23,18,24)(25,28)(26,32)(27,36)(29,38)(30,39)(31,34)(33,40)(35,37);(1,3,5,4)(2,7,6,8)(9,10)(11,18)(12,20)(13,15)(14,17)(16,22)(19,23)(21,24)(25,26)(27,33)(28,31)(29,34)(30,37)(32,38)(35,40)(36,39);(1,4,5,3)(2,8,6,7)(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24)(25,27)(26,33)(28,36)(29,37)(30,34)(31,39)(32,40)(35,38)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7702957);