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 | |
4133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,18,22,20)(19,25,27,23)(21,28,30,24)(26,31,29,32)(33,34,38,36)(35,39,43,41)(37,44,45,40)(42,47,48,46);(2,6)(4,12)(7,10)(11,15)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31)(33,34,38,36)(35,41,43,39)(37,44,45,40)(42,46,48,47);(3,9)(7,10)(11,15)(14,16)(18,20)(23,25)(24,28)(31,32)(34,36)(39,41)(40,44)(46,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gi,b^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10232756);