Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45093 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,12)(9,10)(13,14)(15,20)(16,27)(17,33)(18,34)(19,23)(21,32)(22,29)(24,41)(25,40)(26,36)(28,38)(30,35)(31,43)(37,42)(39,44);(1,2)(3,4)(6,10)(11,12)(13,15,19,26)(14,20,23,36)(16,28,31,30)(17,29,25,18)(21,37,39,41)(22,40,34,33)(24,32,42,44)(27,38,43,35);(5,7,9,8)(6,11,10,12)(13,16,29,44,19,31,18,32)(14,21,34,43,23,39,22,27)(15,24,17,28,26,42,25,30)(20,35,40,37,36,38,33,41);(2,4)(5,8,9,7)(6,12,10,11)(13,17,18,15,19,25,29,26)(14,20,22,33,23,36,34,40)(16,30,32,42,31,28,44,24)(21,38,27,37,39,35,43,41);(6,10)(11,12)(16,31)(21,39)(24,42)(27,43)(28,30)(32,44)(35,38)(37,41)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fg,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469487);