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 | |
4133 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,23,25)(19,27,22,24)(20,21,26,32)(28,29,30,31)(33,34)(35,37)(36,42)(38,43)(39,44)(40,45)(41,48)(46,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,22)(18,24,25,27)(20,28,26,30)(21,31,32,29)(33,35)(34,41)(36,46)(37,45)(38,42)(39,47)(40,48)(43,44);(3,10)(6,7)(11,16)(14,15)(18,25)(20,26)(24,27)(28,30)(33,36,40,44)(34,42,45,39)(35,46,48,43)(37,47,41,38);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20)(18,21)(19,28)(22,30)(23,26)(24,31)(25,32)(27,29)(33,37,40,41)(34,35,45,48)(36,38,44,47)(39,46,42,43);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21)(18,26)(19,29)(20,25)(22,31)(23,32)(24,28)(27,30)(33,38)(34,43)(35,42)(36,37)(39,48)(40,47)(41,44)(45,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,22,23,19)(18,27,25,24)(20,30,26,28)(21,29,32,31)(33,39,40,42)(34,44,45,36)(35,47,48,38)(37,46,41,43)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9074633);