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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,12)(8,10)(9,11)(13,14,19,24)(15,20,28,36)(16,29,31,38)(17,23,18,22)(21,33,39,32)(25,37,42,44)(26,35,27,40)(30,34,41,43);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,28)(14,20,24,36)(16,30,31,41)(17,27,18,26)(21,37,39,44)(22,35,23,40)(25,33,42,32)(29,34,38,43);(13,16,19,31)(14,21,24,39)(15,25,28,42)(17,32,18,33)(20,34,36,43)(22,29,23,38)(26,30,27,41)(35,44,40,37);(2,4)(5,8,9,7)(6,10,11,12)(14,22)(15,26)(16,31)(17,18)(21,38)(23,24)(25,41)(27,28)(29,39)(30,42)(34,43)(35,40);(6,11)(10,12)(13,17,19,18)(14,23,24,22)(15,26,28,27)(16,32,31,33)(20,35,36,40)(21,38,39,29)(25,30,42,41)(34,44,43,37)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425293);