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 | |
4270 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)(17,18,22,24)(19,20,27,23)(21,29,31,25)(26,32,30,28)(33,34,38,39)(35,40,46,41)(36,37,43,42)(44,45,47,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,19,22,27)(18,23,24,20)(21,30,31,26)(25,28,29,32)(34,39)(37,42)(40,41)(45,48);(2,6)(7,10)(8,12)(14,16)(18,24)(20,23)(25,29)(28,32)(33,35,38,46)(34,40,39,41)(36,44,43,47)(37,45,42,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,30)(31,32)(33,35)(34,41)(36,47)(37,45)(38,46)(39,40)(42,48)(43,44);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,32)(28,29)(30,31)(33,36,38,43)(34,42,39,37)(35,44,46,47)(40,48,41,45);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,31)(18,25,24,29)(19,26,27,30)(20,28,23,32)(33,37,38,42)(34,43,39,36)(35,45,46,48)(40,47,41,44)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8762626);