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 | |
4132 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)(4,10)(5,7)(6,9)(11,15)(12,16)(13,14)(17,18)(19,25)(20,27)(21,29)(22,23)(24,31)(26,28)(30,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(2,5)(8,10)(9,13)(15,16)(18,23)(20,25)(24,29)(28,32)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,48)(46,47);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,4,7,3)(2,8,5,10)(6,11,14,12)(9,16,13,15)(17,19)(18,20)(21,30)(22,27)(23,25)(24,32)(26,31)(28,29)(33,34)(35,36)(37,45)(38,41)(39,42)(40,46)(43,48)(44,47);(1,5,7,2)(3,10,4,8)(6,13,14,9)(11,16,12,15)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,30)(31,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,6,7,14)(2,9,5,13)(3,11,4,12)(8,15,10,16)(17,21,22,31)(18,24,23,29)(19,26,27,30)(20,28,25,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|c^2=gh,d^2=gh,e^2=gh,f^2=g,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, 10316889);