Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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×#3561 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3005 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(13,14)(15,24)(16,29)(17,22)(18,32)(19,23)(20,28)(21,30)(25,34)(26,35)(27,38)(31,36)(33,37)(39,40)(41,43)(42,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,23,35)(16,25,31,41)(17,28,18,26)(21,34,37,43)(22,38,32,24)(29,39,36,44)(30,42,33,40);(6,9)(10,12)(13,16,19,31)(14,21,23,37)(15,25,27,41)(17,30,18,33)(20,34,35,43)(22,36,32,29)(24,39,38,44)(26,40,28,42);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,32)(15,26,27,28)(16,30,31,33)(20,24,35,38)(21,36,37,29)(25,40,41,42)(34,39,43,44);(7,11)(10,12)(15,27)(20,35)(24,38)(25,41)(26,28)(34,43)(39,44)(40,42)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449683);