Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #45241 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,21)(17,20)(18,23)(22,29)(24,31)(25,30)(26,33)(27,36)(28,34)(32,39)(35,40)(37,43)(38,41)(42,44);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,26)(14,19,23,33)(16,24,27,37)(17,25,29,39)(20,30,22,32)(21,31,36,43)(28,38,40,44)(34,41,35,42);(5,7,8,12)(6,9,10,11)(14,20)(16,27)(17,29)(19,30)(21,35)(22,23)(24,37)(25,39)(31,42)(32,33)(34,36)(41,43);(1,3,4,2)(7,12)(9,11)(13,16)(14,21)(15,24)(17,28)(18,27)(19,31)(20,34)(22,35)(23,36)(25,38)(26,37)(29,40)(30,41)(32,42)(33,43)(39,44);(6,10)(9,11)(16,27)(21,36)(24,37)(28,40)(31,43)(34,35)(38,44)(41,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6963235);