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 |
| prime radical | #48168 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4371 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)(9,10,14,18)(11,17,19,15)(12,13,16,23)(20,21,24,22)(25,26,31,35)(27,32,30,34)(28,29,39,33)(36,37,38,40);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,20)(13,22)(14,19)(16,24)(17,18)(21,23)(25,27,31,30)(26,32,35,34)(28,36,39,38)(29,37,33,40);(11,19)(15,17)(20,24)(21,22)(25,28,31,39)(26,33,35,29)(27,36,30,38)(32,40,34,37);(1,3,5,4)(2,7,6,8)(9,12)(10,13)(11,20)(14,16)(15,22)(17,21)(18,23)(19,24)(25,28)(26,29)(27,36)(30,38)(31,39)(32,37)(33,35)(34,40);(1,4,5,3)(2,8,6,7)(9,13)(10,16)(11,21)(12,18)(14,23)(15,20)(17,24)(19,22)(25,29)(26,28)(27,37)(30,40)(31,33)(32,36)(34,38)(35,39);(9,11)(10,17)(12,20)(13,21)(14,19)(15,18)(16,24)(22,23)(25,30,31,27)(26,34,35,32)(28,38,39,36)(29,40,33,37)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=h,d^2=gi,e^2=gi,f^2=h,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, 7705263);