Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| 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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4107 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)(13,14,19,24)(15,20,26,35)(16,29,32,37)(17,21,18,23)(22,30,38,31)(25,39,40,43)(27,33,28,36)(34,41,44,42);(6,9)(10,12)(13,15)(14,20)(16,25)(17,27)(18,28)(19,26)(21,33)(22,34)(23,36)(24,35)(29,39)(30,41)(31,42)(32,40)(37,43)(38,44);(2,4)(5,7,8,11)(6,10,9,12)(14,21)(17,18)(20,33)(22,37)(23,24)(27,28)(29,38)(30,31)(34,43)(35,36)(39,44)(41,42);(13,16,19,32)(14,22,24,38)(15,25,26,40)(17,31,18,30)(20,34,35,44)(21,37,23,29)(27,42,28,41)(33,43,36,39);(15,26)(20,35)(25,40)(27,28)(33,36)(34,44)(39,43)(41,42)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,h^a=hi,e^b=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536426);