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 | |
4159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,5,6)(4,9,13,16)(11,15,14,12)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(1,3,7,5)(2,8,10,6)(4,11,13,14)(9,15,16,12)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(3,5)(6,8)(11,14)(12,15)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,4,7,13)(2,9,10,16)(3,11,5,14)(6,12,8,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,5,7,3)(2,6,10,8)(4,11,13,14)(9,15,16,12)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,6)(2,3)(4,12)(5,10)(7,8)(9,11)(13,15)(14,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=hi,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9452825);