Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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×#3516 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3122 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)(5,6)(7,10)(8,9)(11,12)(13,14)(15,27)(16,22)(17,23)(18,26)(19,25)(20,24)(21,30)(28,41)(29,44)(31,39)(32,40)(33,37)(34,43)(35,42)(36,38);(2,4)(13,15,20,32)(14,21,24,39)(16,33,17,36)(18,31,19,30)(22,41,23,44)(25,40,26,27)(28,34,29,35)(37,42,38,43);(6,9)(10,12)(13,16,20,17)(14,22,24,23)(15,28,32,29)(18,35,19,34)(21,37,39,38)(25,43,26,42)(27,41,40,44)(30,33,31,36);(13,17,20,16)(14,23,24,22)(15,29,32,28)(18,34,19,35)(21,38,39,37)(25,42,26,43)(27,44,40,41)(30,36,31,33);(5,7,8,11)(6,10,9,12)(14,24)(21,39)(22,23)(25,26)(27,40)(37,38)(41,44)(42,43)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361136);