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 | |
4165 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,25)(19,24,27,31)(20,22,29,32)(21,26,28,30)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,27)(18,24,25,31)(20,28,29,21)(22,30,32,26)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,3)(7,9)(8,10)(14,15)(18,25)(19,27)(20,29)(26,30)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,22)(19,28)(21,27)(23,29)(24,30)(25,32)(26,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,28)(18,26,25,30)(19,29,27,20)(22,24,32,31)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22)(18,20)(19,30)(21,31)(23,32)(24,28)(25,29)(26,27)(33,36)(34,39)(35,41)(37,44)(38,45)(40,43)(42,48)(46,47)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=hi,d^2=g,e^2=gi,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, 10065828);