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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1736 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)(5,6,9,7)(8,11,12,10)(13,14,19,25)(15,26,30,21)(16,23,17,22)(18,34,20,24)(27,36,31,40)(28,37,44,43)(29,38,35,39)(32,41,33,42);(1,2,4,3)(5,7,9,6)(8,10,12,11)(13,15,19,30)(14,21,25,26)(16,31,17,27)(18,29,20,35)(22,40,23,36)(24,38,34,39)(28,32,44,33)(37,42,43,41);(5,8,9,12)(6,10,7,11)(13,16,19,17)(14,22,25,23)(15,27,30,31)(18,33,20,32)(21,36,26,40)(24,42,34,41)(28,29,44,35)(37,39,43,38);(6,7)(10,11)(13,17,19,16)(14,23,25,22)(15,28,30,44)(18,33,20,32)(21,37,26,43)(24,42,34,41)(27,29,31,35)(36,39,40,38);(1,3,4,2)(13,18,19,20)(14,24,25,34)(15,29,30,35)(16,32,17,33)(21,38,26,39)(22,41,23,42)(27,28,31,44)(36,43,40,37)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,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, 7453175);