Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470161 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3515 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(10,12)(13,14,17,24,19,23,18,22)(15,25,29,37,28,36,27,20)(16,30,33,21,34,39,32,40)(26,38,42,41,31,35,43,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,28)(14,20,23,37)(16,31,34,26)(17,27,18,29)(21,38,40,35)(22,25,24,36)(30,41,39,44)(32,43,33,42);(6,9)(10,12)(13,16,19,34)(14,21,23,40)(15,26,28,31)(17,32,18,33)(20,35,37,38)(22,39,24,30)(25,41,36,44)(27,42,29,43);(6,9)(10,12)(13,17,19,18)(14,22,23,24)(15,27,28,29)(16,32,34,33)(20,25,37,36)(21,39,40,30)(26,42,31,43)(35,41,38,44);(5,7,8,11)(6,10,9,12)(14,23)(15,28)(21,40)(22,24)(26,31)(27,29)(30,39)(42,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398604);