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 | |
3986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,10,5,8)(4,12,13,9)(11,15,14,16)(17,18,23,24)(19,28,29,25)(20,32,26,21)(22,27,30,31)(33,34,38,40)(35,41,42,36)(37,45,46,39)(43,44,48,47);(2,7)(4,13)(8,10)(11,14)(18,24)(19,29)(20,26)(27,31)(35,42)(36,41)(37,46)(39,45);(1,3,6,5)(2,8,7,10)(4,11,13,14)(9,15,12,16)(17,19)(18,25)(20,30)(21,31)(22,26)(23,29)(24,28)(27,32)(33,35)(34,36)(37,43)(38,42)(39,44)(40,41)(45,47)(46,48);(1,4,6,13)(2,9,7,12)(3,11,5,14)(8,15,10,16)(17,20,23,26)(18,21,24,32)(19,30,29,22)(25,31,28,27)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,48)(46,47);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(17,21)(18,26)(19,31)(20,24)(22,25)(23,32)(27,29)(28,30)(33,37,38,46)(34,39,40,45)(35,43,42,48)(36,44,41,47);(1,5,6,3)(2,10,7,8)(4,14,13,11)(9,16,12,15)(17,22,23,30)(18,27,24,31)(19,26,29,20)(21,25,32,28)(33,34)(35,36)(37,39)(38,40)(41,42)(43,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=h,c^2=hi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015927);