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)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,18,22,24)(19,26,30,23)(20,25,27,21)(28,29,31,32)(35,41)(39,46)(42,44)(47,48);(1,3,5,11)(2,6,9,7)(4,10,12,16)(8,13,15,14)(17,19)(18,23)(20,28)(21,29)(22,30)(24,26)(25,32)(27,31)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(4,12)(8,15)(10,16)(13,14)(18,24)(21,25)(23,26)(29,32)(34,38)(39,46)(40,43)(47,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,20,22,27)(18,25,24,21)(19,28,30,31)(23,32,26,29)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19)(18,26)(20,31)(21,29)(22,30)(23,24)(25,32)(27,28)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48);(1,4,5,12)(2,8,9,15)(3,10,11,16)(6,13,7,14)(17,21,22,25)(18,27,24,20)(19,29,30,32)(23,31,26,28)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8832523);