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 | |
3973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,24,29)(19,30,21,25)(20,31,32,26)(22,27,28,23)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,19,24,21)(18,25,29,30)(20,28,32,22)(23,31,27,26)(33,34,38,36)(35,39,43,41)(37,44,45,40)(42,47,48,46);(2,6)(3,9)(4,12)(14,16)(17,20,24,32)(18,26,29,31)(19,28,21,22)(23,30,27,25)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,21,24,19)(18,25,29,30)(20,28,32,22)(23,26,27,31)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,22,24,28)(18,27,29,23)(19,20,21,32)(25,26,30,31)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,23,24,27)(18,28,29,22)(19,31,21,26)(20,25,32,30)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,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, 8964550);