Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| 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 | #37550 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ 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)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,19)(16,26)(17,30)(18,21)(20,29)(22,31)(23,37)(24,33)(25,40)(27,41)(28,42)(32,38)(34,39)(35,43)(36,44);(2,4)(7,8)(10,12)(13,15)(14,19)(16,27)(17,25)(18,24)(20,35)(21,33)(22,34)(23,28)(26,41)(29,43)(30,40)(31,39)(32,36)(37,42)(38,44);(5,7,9,8)(6,10,11,12)(13,16,17,29,18,28,31,44)(14,20,22,26,21,36,30,42)(15,23,25,38,24,27,39,43)(19,32,34,37,33,35,40,41);(5,8,9,7)(6,10,11,12)(15,24)(19,33)(23,27)(25,39)(32,35)(34,40)(37,41)(38,43);(6,11)(10,12)(14,21)(15,24)(16,28)(22,30)(25,39)(29,44)(32,35)(37,41)Polycyclic presentation:
2^9|c^2=g,d^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390344);