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 | |
4150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,10,5,8)(4,6,13,16)(11,15,14,12)(19,25)(24,31)(26,29)(27,32)(33,34,39,41)(35,40,44,46)(36,45,43,38)(37,42,47,48);(1,3,7,5)(2,8,9,10)(4,11,13,14)(6,12,16,15)(17,18,22,23)(19,24,25,31)(20,28,30,21)(26,27,29,32)(33,35)(34,40)(36,42)(37,38)(39,44)(41,46)(43,48)(45,47);(2,9)(6,16)(8,10)(12,15)(18,23)(21,28)(24,31)(27,32)(34,41)(35,44)(38,45)(42,48);(1,4)(2,6)(3,11)(5,14)(7,13)(8,12)(9,16)(10,15)(17,19)(18,24)(20,29)(21,27)(22,25)(23,31)(26,30)(28,32)(33,36,39,43)(34,38,41,45)(35,42,44,48)(37,46,47,40);(1,5,7,3)(2,8,9,10)(4,11,13,14)(6,15,16,12)(17,20)(18,21)(19,26)(22,30)(23,28)(24,32)(25,29)(27,31)(33,37)(34,42)(35,38)(36,40)(39,47)(41,48)(43,46)(44,45);(1,6)(2,4)(3,12)(5,15)(7,16)(8,11)(9,13)(10,14)(17,21)(18,20)(19,27)(22,28)(23,30)(24,29)(25,32)(26,31)(33,38)(34,43)(35,37)(36,41)(39,45)(40,48)(42,46)(44,47)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=gh,e^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, 8833514);