Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469971 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3219 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,19,24)(15,25,27,40)(16,21,31,38)(17,23,18,22)(20,33,35,28)(26,41,29,44)(30,39,32,37)(34,42,36,43);(7,8)(10,12)(13,15)(14,20)(16,29)(17,28)(18,33)(19,27)(21,36)(22,25)(23,40)(24,35)(26,31)(30,43)(32,42)(34,38)(37,41)(39,44);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16)(14,21)(15,26)(17,32)(18,30)(19,31)(20,34)(22,39)(23,37)(24,38)(25,41)(27,29)(28,43)(33,42)(35,36)(40,44);(5,8,9,7)(6,10,11,12)(15,27)(20,35)(25,40)(26,29)(28,33)(34,36)(41,44)(42,43);(6,11)(10,12)(13,17,19,18)(14,22,24,23)(15,28,27,33)(16,30,31,32)(20,25,35,40)(21,37,38,39)(26,42,29,43)(34,44,36,41)Polycyclic presentation:
2^9|a^2=i,c^2=g,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,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390420);