Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,9)(7,11,12,10)(13,14)(15,25)(16,22)(17,23)(18,34)(19,24)(20,29)(21,32)(26,42)(27,40)(28,36)(30,41)(31,38)(33,37)(35,44)(39,43);(2,3)(6,9)(10,11)(13,15,17,28,19,30,18,29)(14,20,34,41,24,36,23,25)(16,26,33,44,32,27,31,43)(21,35,37,42,22,39,38,40);(1,2,4,3)(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,26)(17,33)(18,31)(19,32)(20,35)(22,24)(23,38)(25,40)(27,30)(28,44)(29,43)(34,37)(36,39)(41,42);(1,3,4,2)(13,16)(14,22)(15,27)(17,33)(18,31)(19,32)(20,35)(21,24)(23,37)(25,40)(26,30)(28,43)(29,44)(34,38)(36,39)(41,42);(13,17,19,18)(14,23,24,34)(15,28,30,29)(16,31,32,33)(20,25,36,41)(21,37,22,38)(26,43,27,44)(35,42,39,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=ghi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374066);