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 | |
4338 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)(3,9)(4,8)(5,6)(7,10)(11,16)(12,14)(13,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(2,6)(3,10)(8,14)(11,13)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,35,39,37)(34,40,36,41)(38,45,46,43)(42,48,44,47);(4,12)(8,14)(11,13)(15,16)(18,22)(23,25)(24,28)(30,31)(33,34,39,36)(35,41,37,40)(38,42,46,44)(43,48,45,47);(1,3)(2,7)(4,13)(5,10)(6,9)(8,16)(11,12)(14,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,4)(2,8)(3,11)(5,12)(6,14)(7,15)(9,16)(10,13)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=hi,d^2=hi,e^2=hi,f^2=hi,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, 10315060);