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 | |
4299 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(35,41)(39,46)(42,44)(47,48);(2,6)(7,10)(8,13)(14,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(3,9)(4,12)(7,10)(8,13)(18,20)(22,30)(24,26)(27,29)(34,38)(35,41)(40,43)(42,44);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,20,23,18)(19,26,21,24)(22,28,30,25)(27,32,29,31)(33,35,37,41)(34,39,38,46)(36,44,45,42)(40,48,43,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,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, 8837506);