Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | #47723 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,4,8)(9,10,14,15)(11,18,20,16)(12,17,13,22)(19,24,21,23);(2,6)(7,8)(10,15)(16,18)(17,22)(23,24)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(9,11)(10,16)(12,21)(13,19)(14,20)(15,18)(17,24)(22,23)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,15,22)(11,19,20,21)(16,23,18,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,17,15,22)(11,19,20,21)(16,24,18,23)(27,35)(30,33)(34,40)(38,39);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609033);