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 | |
4143 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,16)(13,15)(17,18,23,24)(19,22,27,30)(20,25,29,31)(21,26,28,32)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(2,7)(3,11)(5,14)(12,15)(17,19,23,27)(18,22,24,30)(20,28,29,21)(25,32,31,26)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(2,7)(4,9)(8,10)(12,15)(18,24)(19,27)(21,28)(25,31)(37,45)(40,44)(42,48)(46,47);(1,3,6,11)(2,8,7,10)(4,12,9,15)(5,13,14,16)(17,20)(18,25)(19,21)(22,26)(23,29)(24,31)(27,28)(30,32)(33,36,38,34)(35,39,43,41)(37,40,45,44)(42,47,48,46);(1,4,6,9)(2,5,7,14)(3,12,11,15)(8,13,10,16)(17,21)(18,26)(19,20)(22,25)(23,28)(24,32)(27,29)(30,31)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,12)(11,16)(17,22)(18,19)(20,26)(21,25)(23,30)(24,27)(28,31)(29,32)(33,35)(34,41)(36,39)(37,42)(38,43)(40,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=g,e^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, 10231757);