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 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18)(19,26)(20,24)(21,29)(22,23)(25,30)(27,28)(31,32)(33,34,39,43)(35,38,45,47)(36,41,46,40)(37,42,44,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,22,23)(19,26,24,20)(21,25,30,29)(27,32,31,28)(33,35)(34,38)(36,44)(37,46)(39,45)(40,42)(41,48)(43,47);(2,6)(7,9)(8,13)(14,15)(18,23)(19,24)(25,29)(27,31)(33,36,39,46)(34,40,43,41)(35,44,45,37)(38,42,47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,22,24)(18,20,23,26)(21,27,30,31)(25,28,29,32)(33,36)(34,41)(35,37)(38,42)(39,46)(40,43)(44,45)(47,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,29)(30,32)(33,37)(34,42)(35,36)(38,41)(39,44)(40,47)(43,48)(45,46);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,38)(34,35)(36,42)(37,41)(39,47)(40,44)(43,45)(46,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000611);