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 | |
4234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,10,8)(4,12,15,9)(11,16,13,14)(17,18,22,25)(19,26,20,23)(21,24,29,31)(27,32,28,30)(33,34,39,42)(35,38,41,47)(36,37,40,46)(43,44,45,48);(2,7)(3,10)(9,12)(11,13)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,35)(34,38)(36,43)(37,44)(39,41)(40,45)(42,47)(46,48);(2,7)(5,8)(9,12)(14,16)(21,29)(24,31)(27,28)(30,32)(35,41)(38,47)(43,45)(44,48);(1,3,6,10)(2,8,7,5)(4,13,15,11)(9,14,12,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,32)(29,31)(33,36)(34,37)(35,43)(38,44)(39,40)(41,45)(42,46)(47,48);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,37)(34,40)(35,44)(36,42)(38,45)(39,46)(41,48)(43,47);(1,5,6,8)(2,10,7,3)(4,14,15,16)(9,13,12,11)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,38,39,47)(34,41,42,35)(36,44,40,48)(37,45,46,43)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,d^2=h,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8231548);