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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,32)(21,37)(22,27)(23,28)(25,35)(26,31)(29,41)(30,40)(33,39)(34,42)(36,47)(38,44)(43,48)(45,46);(2,6)(7,8)(11,16)(13,15)(17,19,22,30)(18,24,27,40)(20,33,35,29)(21,38,31,23)(25,41,32,39)(26,28,37,44)(34,36,45,48)(42,47,46,43);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,36)(24,39)(26,42)(27,32)(28,43)(30,33)(31,45)(37,46)(38,48)(40,41)(44,47);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,21,23,19,22,31,38,30)(18,26,28,40,27,37,44,24)(20,34,36,29,35,45,48,33)(25,42,43,41,32,46,47,39);(2,6)(7,8)(20,35)(25,32)(29,33)(34,45)(36,48)(39,41)(42,46)(43,47)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=fgi,d^2=fh,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=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7522098);