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 | |
4208 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(18,24)(19,20)(26,29)(27,28)(35,39)(36,42)(43,45)(44,47);(3,12)(6,15)(8,16)(10,13)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,35,38,39)(34,36,41,42)(37,45,46,43)(40,47,48,44);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,45)(43,48)(46,47);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,37)(34,40)(35,43)(36,44)(38,46)(39,45)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8313879);