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 | |
4117 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)(3,10,9,5)(4,12,14,8)(11,13,15,16)(19,25)(23,26)(27,31)(30,32)(33,34,38,39)(35,36,41,44)(37,45,47,40)(42,48,46,43);(2,7)(5,10)(8,12)(13,16)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(33,35)(34,36)(37,46)(38,41)(39,44)(40,48)(42,47)(43,45);(2,7)(3,9)(8,12)(11,15)(18,22)(23,26)(24,28)(30,32)(34,39)(35,41)(40,45)(42,46);(1,3,6,9)(2,5,7,10)(4,11,14,15)(8,13,12,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,48)(46,47);(1,4,6,14)(2,8,7,12)(3,11,9,15)(5,13,10,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,46)(47,48);(1,5)(2,9)(3,7)(4,13)(6,10)(8,15)(11,12)(14,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37,38,47)(34,40,39,45)(35,42,41,46)(36,43,44,48)Polycyclic presentation:
2^9|a^2=h,d^2=g,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8663257);