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 | #43021 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4541 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,17)(12,16)(13,18)(14,15)(19,23)(20,22)(21,24)(25,26,31,29)(27,35,36,32)(28,33,37,38)(30,34,40,39);(1,2,5,6)(3,7,4,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,24)(22,23)(25,27,31,36)(26,32,29,35)(28,30,37,40)(33,39,38,34);(10,15)(11,18)(12,20)(21,23)(27,36)(30,40)(32,35)(34,39);(1,3,5,4)(2,7,6,8)(9,12,14,20)(10,16,15,22)(11,19,18,24)(13,21,17,23)(25,28,31,37)(26,33,29,38)(27,30,36,40)(32,39,35,34);(1,4,5,3)(2,8,6,7)(9,13,14,17)(10,11,15,18)(12,21,20,23)(16,19,22,24)(25,29,31,26)(27,32,36,35)(28,38,37,33)(30,39,40,34);(9,11)(10,17)(12,19)(13,15)(14,18)(16,23)(20,24)(21,22)(25,30)(26,34)(27,37)(28,36)(29,39)(31,40)(32,33)(35,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7652531);