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 | |
4309 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,10,4,7)(5,13,14,8)(11,15,12,16)(17,18,23,28)(19,24,30,32)(20,21,26,25)(22,31,29,27)(33,34)(35,37)(36,40)(38,41)(39,42)(43,45)(44,46)(47,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,23,30)(18,24,28,32)(20,29,26,22)(21,27,25,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,20)(18,25)(19,29)(21,28)(22,30)(23,26)(24,31)(27,32)(33,36,39,37)(34,40,42,35)(38,44,47,45)(41,46,48,43);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,20)(18,21)(19,29)(22,30)(23,26)(24,27)(25,28)(31,32)(33,35)(34,36)(37,42)(38,46)(39,40)(41,45)(43,47)(44,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,21)(18,26)(19,27)(20,28)(22,24)(23,25)(29,32)(30,31)(33,37,39,36)(34,40,42,35)(38,45,47,44)(41,46,48,43);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22,23,29)(18,27,28,31)(19,20,30,26)(21,24,25,32)(33,38,39,47)(34,41,42,48)(35,43,40,46)(36,44,37,45)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10331801);