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 | |
4119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,5,8,10)(6,9,15,12)(11,13,14,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(33,34,40,44)(35,45,48,41)(36,37,38,42)(39,46,47,43);(1,3,7,8)(2,5,4,10)(6,14,15,11)(9,16,12,13)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(33,35,40,48)(34,41,44,45)(36,43,38,46)(37,47,42,39);(1,2,7,4)(3,10,8,5)(6,9,15,12)(11,16,14,13)(17,18,21,24)(19,22,27,25)(20,23,29,28)(26,30,32,31)(33,36,40,38)(34,42,44,37)(35,46,48,43)(39,41,47,45);(1,4,7,2)(3,5,8,10)(6,12,15,9)(11,13,14,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,37)(34,36)(35,47)(38,44)(39,48)(40,42)(41,43)(45,46);(1,5)(2,8)(3,4)(6,13)(7,10)(9,14)(11,12)(15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,38,40,36)(34,42,44,37)(35,46,48,43)(39,45,47,41);(1,6)(2,9)(3,11)(4,12)(5,13)(7,15)(8,14)(10,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)(33,39,40,47)(34,43,44,46)(35,37,48,42)(36,41,38,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=i,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10275848);