Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45134 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,26)(20,34)(21,36)(22,28)(23,30)(24,29)(25,32)(27,39)(31,44)(33,42)(35,41)(37,43)(38,46)(40,45)(47,48);(2,6)(7,8)(10,13)(11,16)(17,19,22,32)(18,25,28,26)(20,24,33,23)(21,35,38,37)(27,44,45,48)(29,34,30,42)(31,39,47,40)(36,43,46,41);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,23,19,22,33,24,32)(18,26,29,34,28,25,30,42)(21,37,39,47,38,35,40,31)(27,41,36,48,45,43,46,44);(1,4,5,3)(2,8,6,7)(10,13)(14,15)(17,21,23,39,22,38,24,40)(18,27,29,36,28,45,30,46)(19,31,33,37,32,47,20,35)(25,41,42,48,26,43,34,44);(2,6)(7,8)(10,13)(14,15)(18,28)(21,38)(25,26)(29,30)(31,47)(34,42)(35,37)(39,40)Polycyclic presentation:
2^9|b^2=i,c^2=fh,d^2=fgi,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,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7503187);