Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×#3524 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3128 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)(15,25)(16,21)(17,22)(18,23)(19,24)(20,26)(27,41)(28,36)(29,35)(30,44)(31,38)(32,39)(33,40)(34,42)(37,43);(2,4)(13,15,19,29)(14,20,24,36)(16,30,33,42)(17,28,18,26)(21,37,40,41)(22,25,23,35)(27,31,43,32)(34,39,44,38);(6,9)(10,12)(15,26)(17,18)(20,25)(22,23)(27,42)(28,29)(30,43)(31,32)(34,41)(35,36)(37,44)(38,39);(13,16,19,33)(14,21,24,40)(15,27,29,43)(17,31,18,32)(20,34,36,44)(22,38,23,39)(25,41,35,37)(26,42,28,30);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,24,23)(15,28,29,26)(16,31,33,32)(20,25,36,35)(21,38,40,39)(27,30,43,42)(34,41,44,37)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361300);