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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4397 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,9)(6,7)(8,11)(10,13)(12,14)(15,16)(17,18,22,23)(19,25,27,20)(21,29,30,24)(26,28,32,31)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(2,7)(5,8)(9,14)(13,15)(18,23)(20,25)(21,30)(26,32)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(4,12)(9,14)(10,16)(13,15)(17,19,22,27)(18,20,23,25)(21,26,30,32)(24,28,29,31)(33,34,37,40)(35,41,43,38)(36,39,46,44)(42,48,45,47);(1,3)(2,8)(4,10)(5,7)(6,11)(9,15)(12,16)(13,14)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,47)(44,46)(45,48);(1,4)(2,9)(3,10)(5,13)(6,12)(7,14)(8,15)(11,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,35)(34,38)(36,42)(37,43)(39,47)(40,41)(44,48)(45,46);(1,5,6,8)(2,3,7,11)(4,13,12,15)(9,10,14,16)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)(33,36)(34,39)(35,42)(37,46)(38,47)(40,44)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150490);