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 | #39120 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,26)(27,33)(28,38)(29,37)(30,35)(31,32)(34,39)(36,40);(3,7)(6,8)(11,12)(13,21)(15,18)(16,23)(26,32)(29,39)(33,38)(35,36);(1,2,5,4)(3,6,7,8)(9,11,14,12)(10,15,17,18)(13,19,21,20)(16,22,23,24)(25,27,31,28)(26,33,32,38)(29,36,39,35)(30,37,40,34);(1,3,5,7)(2,6,4,8)(9,12,14,11)(10,15,17,18)(13,20,21,19)(16,22,23,24)(25,28,31,27)(26,33,32,38)(29,35,39,36)(30,37,40,34);(1,4,5,2)(3,8,7,6)(9,13,14,21)(10,16,17,23)(11,19,12,20)(15,22,18,24)(25,29)(26,34)(27,36)(28,35)(30,33)(31,39)(32,37)(38,40);(9,10)(11,15)(12,18)(13,16)(14,17)(19,22)(20,24)(21,23)(25,30)(26,35)(27,37)(28,34)(29,33)(31,40)(32,36)(38,39)Polycyclic presentation:
2^9|c^2=gh,d^2=gh,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, 8380153);