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 | #36738 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4490 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)(10,15)(13,23)(16,24)(17,20)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,20)(13,18)(14,15)(17,22)(19,23)(21,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(10,15)(12,22)(16,24)(18,19)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(1,3,4,7)(2,6,5,8)(9,11,14,21)(10,16,15,24)(12,19,22,18)(13,20,23,17)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(9,12)(10,17)(11,19)(13,16)(14,22)(15,20)(18,21)(23,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(9,13)(10,18)(11,20)(12,16)(14,23)(15,19)(17,21)(22,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7623051);