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 | #40780 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,17)(12,19)(13,16)(14,15)(18,24)(20,21)(22,23)(25,26,30,34)(27,29,33,31)(28,32,38,40)(35,37,36,39);(2,5)(6,7)(10,15)(12,17)(16,23)(20,24)(25,27,30,33)(26,29,34,31)(28,36,38,35)(32,39,40,37);(1,2)(3,6)(4,5)(7,8)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,21)(14,19)(15,17)(16,24)(18,22)(20,23)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,38)(39,40);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,28,30,38)(26,32,34,40)(27,35,33,36)(29,37,31,39);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,29)(26,33)(27,34)(28,37)(30,31)(32,36)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8379592);