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 | |
3942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,8,6)(3,9,14,13)(4,15,5,10)(7,16,12,11)(18,23)(20,24)(25,31)(28,29)(33,34,39,42)(35,40,45,48)(36,37,38,41)(43,44,46,47);(1,3,8,14)(2,9,6,13)(4,12,5,7)(10,16,15,11)(17,18,22,23)(19,24,27,20)(21,25,30,31)(26,29,32,28)(33,35)(34,40)(36,46)(37,47)(38,43)(39,45)(41,44)(42,48);(1,4,8,5)(2,10,6,15)(3,12,14,7)(9,16,13,11)(17,19,22,27)(18,24,23,20)(21,26,30,32)(25,29,31,28)(33,36,39,38)(34,41,42,37)(35,43,45,46)(40,47,48,44);(1,5,8,4)(2,10,6,15)(3,7,14,12)(9,16,13,11)(17,20)(18,19)(21,29)(22,24)(23,27)(25,32)(26,31)(28,30)(33,35)(34,40)(36,43)(37,44)(38,46)(39,45)(41,47)(42,48);(1,6,8,2)(3,13,14,9)(4,15,5,10)(7,16,12,11)(17,18)(19,24)(20,27)(21,25)(22,23)(26,29)(28,32)(30,31)(33,37)(34,36)(35,44)(38,42)(39,41)(40,43)(45,47)(46,48);(1,7)(2,11)(3,4)(5,14)(6,16)(8,12)(9,10)(13,15)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,29)(27,32)(33,38,39,36)(34,37,42,41)(35,43,45,46)(40,47,48,44)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10358796);