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 | |
4002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,10,11)(5,8,13,12)(9,14,15,16)(19,27)(20,29)(22,28)(24,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(2,4)(7,11)(8,12)(14,16)(17,18,23,26)(19,28,27,22)(20,24,29,32)(21,25,31,30)(36,44)(39,46)(42,48)(45,47);(5,13)(8,12)(9,15)(14,16)(17,19,23,27)(18,22,26,28)(20,21,29,31)(24,30,32,25)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,3,6,10)(2,7,4,11)(5,9,13,15)(8,14,12,16)(17,20)(18,24)(19,21)(22,30)(23,29)(25,28)(26,32)(27,31);(1,4,6,2)(3,7,10,11)(5,12,13,8)(9,14,15,16)(17,21)(18,25)(19,29)(20,27)(22,24)(23,31)(26,30)(28,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48);(1,5)(2,8)(3,9)(4,12)(6,13)(7,14)(10,15)(11,16)(17,22)(18,19)(20,30)(21,24)(23,28)(25,29)(26,27)(31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,d^2=g,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, 8833731);