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 | |
4214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,10,8,4)(5,13,14,9)(11,12,15,16)(17,18)(19,25)(20,28)(21,24)(22,26)(23,30)(27,31)(29,32)(35,41)(38,42)(43,47)(46,48);(2,7)(4,10)(5,14)(11,15)(17,18,21,24)(19,22,26,25)(20,23,30,28)(27,31,29,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,42)(43,48)(46,47);(2,7)(3,8)(9,13)(11,15)(19,26)(22,25)(27,29)(31,32)(33,34,37,40)(35,38,41,42)(36,39,45,44)(43,46,47,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,19)(18,22)(20,29)(21,26)(23,32)(24,25)(27,30)(28,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,45)(46,47);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,20)(18,23)(19,27)(21,30)(22,31)(24,28)(25,32)(26,29)(33,35)(34,38)(36,43)(37,41)(39,46)(40,42)(44,48)(45,47);(1,5,6,14)(2,9,7,13)(3,11,8,15)(4,12,10,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,31)(28,30)(29,32)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8738319);