Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4283 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,18,20)(12,16,19,22)(17,21,23,24);(2,5)(6,8)(9,11,13,18)(10,14,15,20)(12,17,19,23)(16,21,22,24)(25,26)(27,34)(28,38)(29,32)(30,33)(31,37)(35,39)(36,40);(10,15)(12,19)(14,20)(17,23)(25,27,30,37)(26,31,33,34)(28,36,29,35)(32,40,38,39);(9,12,13,19)(10,16,15,22)(11,17,18,23)(14,21,20,24)(25,28,30,29)(26,32,33,38)(27,35,37,36)(31,39,34,40);(10,15)(14,20)(16,22)(21,24)(25,29,30,28)(26,32,33,38)(27,36,37,35)(31,39,34,40);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615367);