Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628847 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4121 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)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,29)(17,21)(18,24)(19,23)(22,31)(25,35)(26,40)(27,36)(28,33)(30,38)(32,37)(34,42)(39,44)(41,43);(6,9)(10,12)(13,15)(14,20)(16,26)(17,27)(18,28)(19,25)(21,36)(22,34)(23,35)(24,33)(29,40)(30,41)(31,42)(32,39)(37,44)(38,43);(2,4)(5,7,8,11)(6,10,9,12)(14,21)(15,25)(17,18)(20,33)(22,37)(23,24)(26,39)(29,38)(30,31)(34,40)(35,36)(43,44);(13,16,19,32)(14,22,23,38)(15,26,25,39)(17,31,18,30)(20,34,35,43)(21,37,24,29)(27,42,28,41)(33,40,36,44);(14,23)(20,35)(21,24)(22,38)(29,37)(33,36)(34,43)(40,44)Polycyclic presentation:
2^9|c^2=f,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=ci,g^c=gi,h^c=hi,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536420);