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 | |
4080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,9,4)(5,10,14,15)(11,13,16,12)(19,25)(24,31)(26,29)(27,32)(33,34)(35,38)(36,41)(37,42)(39,43)(40,45)(44,46)(47,48);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,13,15,12)(17,18,22,23)(19,24,25,31)(20,28,30,21)(26,27,29,32)(33,35)(34,40)(36,44)(37,41)(38,43)(39,45)(42,47)(46,48);(2,8)(3,9)(10,15)(11,16)(18,23)(21,28)(24,31)(27,32)(33,36,39,47)(34,41,43,48)(35,44,45,42)(37,38,46,40);(1,4,6,7)(2,9,8,3)(5,13,14,12)(10,11,15,16)(17,19)(18,24)(20,29)(21,27)(22,25)(23,31)(26,30)(28,32)(33,37)(34,42)(35,41)(36,38)(39,46)(40,47)(43,44)(45,48);(1,5,6,14)(2,10,8,15)(3,11,9,16)(4,12,7,13)(17,20)(18,21)(19,26)(22,30)(23,28)(24,32)(25,29)(27,31)(33,38)(34,35)(36,46)(37,47)(39,40)(41,44)(42,48)(43,45);(1,2)(3,7)(4,9)(5,10)(6,8)(11,13)(12,16)(14,15)(17,21)(18,20)(19,27)(22,28)(23,30)(24,29)(25,32)(26,31)(33,35)(34,38)(36,42)(37,41)(39,45)(40,43)(44,47)(46,48)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=h,e^2=h,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, 8833584);