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 | |
4315 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)(3,7)(4,5)(6,8)(9,11)(10,15)(12,13)(14,16)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(33,34,38,42)(35,41,45,39)(36,40,46,37)(43,44,47,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,9)(10,14)(15,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,38,45)(34,39,42,41)(36,43,46,47)(37,44,40,48);(2,8)(3,11)(5,15)(12,14)(18,20)(22,30)(24,26)(27,29)(36,46)(37,40)(43,47)(44,48);(1,3)(2,9)(4,14)(5,13)(6,11)(7,8)(10,12)(15,16)(17,20,23,18)(19,26,21,24)(22,28,30,25)(27,32,29,31)(33,36)(34,40)(35,43)(37,42)(38,46)(39,48)(41,44)(45,47);(1,4,6,10)(2,5,8,15)(3,12,11,14)(7,13,9,16)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,35)(34,41)(36,47)(37,44)(38,45)(39,42)(40,48)(43,46);(1,5)(2,10)(3,13)(4,8)(6,15)(7,14)(9,12)(11,16)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,37,38,40)(34,36,42,46)(35,44,45,48)(39,43,41,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8837408);