Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469971 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1692 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18,22,25)(19,27,29,42)(20,24,32,41)(21,34,36,26)(23,35,38,30)(28,43,31,45)(33,48,47,40)(37,46,39,44);(11,12)(14,15)(17,19)(18,23)(20,31)(21,35)(22,29)(24,39)(25,38)(26,42)(27,34)(28,32)(30,36)(33,44)(37,41)(40,43)(45,48)(46,47);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,41)(26,40)(27,43)(29,31)(30,44)(34,48)(35,46)(36,47)(38,39)(42,45);(9,12,13,11)(10,15,16,14)(19,29)(23,38)(27,42)(28,31)(30,35)(37,39)(43,45)(44,46);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(18,25)(20,32)(23,38)(26,34)(27,42)(28,31)(33,47)(44,46)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396227);