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 | |
4074 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)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,19)(20,23,28,25)(21,26,30,24)(27,32,29,31)(33,34,39,44)(35,43,38,40)(36,37,42,41)(45,46,47,48);(3,9)(7,13)(10,15)(14,16)(17,18,22,19)(20,25,28,23)(21,24,30,26)(27,32,29,31)(33,35,39,38)(34,40,44,43)(36,45,42,47)(37,48,41,46);(2,6)(7,13)(8,11)(14,16)(18,19)(23,25)(24,26)(31,32)(33,36,39,42)(34,41,44,37)(35,45,38,47)(40,46,43,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32)(33,36)(34,41)(35,45)(37,44)(38,47)(39,42)(40,46)(43,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20)(18,23)(19,25)(21,29)(22,28)(24,32)(26,31)(27,30)(33,37,39,41)(34,42,44,36)(35,46,38,48)(40,45,43,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,38,39,35)(34,43,44,40)(36,47,42,45)(37,46,41,48)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=g,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10034846);