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 | #41404 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,12)(13,17)(14,15)(16,18)(19,20)(21,22)(23,24)(25,26,31,28)(27,35,36,32)(29,33,40,37)(30,34,39,38);(1,2)(3,6)(4,5)(7,8)(9,10,14,15)(11,18,16,12)(13,21,22,17)(19,20,23,24)(25,27)(26,32)(28,35)(29,39)(30,40)(31,36)(33,34)(37,38);(10,15)(11,16)(13,22)(20,24)(27,36)(30,39)(32,35)(34,38);(1,3,4,7)(2,6,5,8)(9,11,14,16)(10,12,15,18)(13,19,22,23)(17,20,21,24)(25,28,31,26)(27,32,36,35)(29,37,40,33)(30,38,39,34);(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,21)(22,24)(25,29)(26,33)(27,30)(28,37)(31,40)(32,38)(34,35)(36,39);(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,30)(26,34)(27,29)(28,38)(31,39)(32,37)(33,35)(36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7712253);