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 | #47990 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(10,16)(13,14)(17,18,22,26)(19,25,21,23)(20,28,30,24)(27,31,29,32)(33,34,39,43)(35,44,45,40)(36,37,41,46)(38,47,48,42);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,19,22,21)(18,23,26,25)(20,29,30,27)(24,32,28,31)(33,35)(34,40)(36,38)(37,42)(39,45)(41,48)(43,44)(46,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,19,22,21)(18,23,26,25)(20,27,30,29)(24,31,28,32)(33,36,39,41)(34,37,43,46)(35,38,45,48)(40,42,44,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28)(33,34)(35,40)(36,37)(38,42)(39,43)(41,46)(44,45)(47,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,21,22,19)(18,25,26,23)(20,27,30,29)(24,31,28,32)(33,37)(34,41)(35,42)(36,43)(38,44)(39,46)(40,48)(45,47);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,30)(29,32)(33,38,39,48)(34,42,43,47)(35,36,45,41)(37,44,46,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044306);