Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3929 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,13,16)(4,5,9,14)(6,15,12,10)(18,22)(19,25)(20,28)(30,32)(33,34,39,37)(35,42,38,40)(36,44,46,41)(43,47,45,48);(1,3,7,13)(2,8,11,16)(4,12,9,6)(5,10,14,15)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,35,39,38)(34,40,37,42)(36,43,46,45)(41,47,44,48);(1,4,7,9)(2,5,11,14)(3,6,13,12)(8,15,16,10)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,34,39,37)(35,40,38,42)(36,44,46,41)(43,48,45,47);(1,4)(2,5)(3,12)(6,13)(7,9)(8,10)(11,14)(15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,39,46)(34,41,37,44)(35,43,38,45)(40,47,42,48);(1,5)(2,9)(3,10)(4,11)(6,8)(7,14)(12,16)(13,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,37,39,34)(35,42,38,40)(36,41,46,44)(43,47,45,48);(1,6,7,12)(2,10,11,15)(3,4,13,9)(5,8,14,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,38,39,35)(34,42,37,40)(36,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8081592);