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 | |
3971 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,34,38,42)(35,40,36,39)(37,41,46,45)(43,48,44,47);(2,7)(3,10)(9,12)(11,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,48)(44,47);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)(33,36,38,35)(34,40,42,39)(37,44,46,43)(41,48,45,47);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)Polycyclic presentation:
2^9|a^2=hi,c^2=gh,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8094114);