Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6,9,10)(7,11,8,12)(13,14,17,21)(15,23,26,19)(16,28,30,20)(18,22,32,38)(24,33,29,36)(25,35,27,34)(31,41,42,37)(39,43,40,44);(2,3)(6,10)(11,12)(13,15,17,26)(14,19,21,23)(16,29,30,24)(18,25,32,27)(20,36,28,33)(22,34,38,35)(31,40,42,39)(37,44,41,43);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,17,30)(14,20,21,28)(15,24,26,29)(18,31,32,42)(19,33,23,36)(22,37,38,41)(25,39,27,40)(34,43,35,44);(1,3,4,2)(5,7,9,8)(6,11,10,12)(15,25)(18,32)(19,34)(22,38)(23,35)(24,39)(26,27)(29,40)(31,42)(33,43)(36,44)(37,41);(5,8,9,7)(6,11,10,12)(14,21)(15,26)(16,30)(22,38)(25,27)(31,42)(33,36)(43,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=fg,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7473219);