Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ 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:
(5,6)(7,11)(8,10)(9,12)(13,14,18,23)(15,25,27,20)(16,30,32,38)(17,24,19,22)(21,33,37,31)(26,40,41,44)(28,35,29,36)(34,42,39,43);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,18,27)(14,20,23,25)(16,26,32,41)(17,28,19,29)(21,34,37,39)(22,35,24,36)(30,44,38,40)(31,42,33,43);(2,4)(13,16,18,32)(14,21,23,37)(15,26,27,41)(17,31,19,33)(20,34,25,39)(22,30,24,38)(28,42,29,43)(35,44,36,40);(1,2)(3,4)(5,7,9,8)(6,11,12,10)(15,27)(20,25)(26,41)(28,29)(34,39)(35,36)(40,44)(42,43);(2,4)(5,8,9,7)(6,10,12,11)(13,17,18,19)(14,22,23,24)(15,28,27,29)(16,31,32,33)(20,35,25,36)(21,30,37,38)(26,42,41,43)(34,44,39,40)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486028);