Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,21)(17,20)(18,23)(22,29)(24,31)(25,32)(26,30)(27,35)(28,34)(33,39)(36,40)(37,41)(38,43)(42,44);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15,18,25)(14,19,23,32)(16,24,27,38)(17,26,29,39)(20,30,22,33)(21,31,35,43)(28,37,40,44)(34,41,36,42);(1,3,4,2)(7,11)(9,12)(14,20)(17,29)(19,30)(21,34)(22,23)(26,39)(28,40)(31,41)(32,33)(35,36)(37,44)(42,43);(7,11)(9,12)(13,16)(14,21)(15,24)(17,28)(18,27)(19,31)(20,34)(22,36)(23,35)(25,38)(26,37)(29,40)(30,41)(32,43)(33,42)(39,44);(6,10)(9,12)(15,25)(16,27)(19,32)(21,35)(26,39)(28,40)(30,33)(34,36)Polycyclic presentation:
2^9|b^2=fi,c^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377978);