Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470374 |
| 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 | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1983 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)(8,11)(9,12)(13,14,19,24)(15,25,27,36)(16,21,32,39)(17,23,18,22)(20,28,35,29)(26,41,30,44)(31,40,33,38)(34,43,37,42);(7,8)(10,11)(13,15,18,29,19,27,17,28)(14,20,23,25,24,35,22,36)(16,30,31,43,32,26,33,42)(21,37,38,44,39,34,40,41);(2,4)(5,7,9,8)(6,10,12,11)(13,16,19,32)(14,21,24,39)(15,26,27,30)(17,33,18,31)(20,34,35,37)(22,40,23,38)(25,41,36,44)(28,43,29,42);(5,8,9,7)(6,11,12,10)(15,27)(20,35)(25,36)(26,30)(28,29)(34,37)(41,44)(42,43);(6,12)(10,11)(13,17,19,18)(14,22,24,23)(15,28,27,29)(16,31,32,33)(20,36,35,25)(21,38,39,40)(26,42,30,43)(34,41,37,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6395031);