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 | |
4229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,9,14,13)(5,15,16,10)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31)(33,34,38,41)(35,39,44,47)(36,37,45,40)(42,43,48,46);(2,7)(4,14)(8,11)(10,15)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31)(33,35,38,44)(34,39,41,47)(36,42,45,48)(37,43,40,46);(2,7)(3,12)(9,13)(10,15)(18,23)(19,26)(24,29)(27,32)(33,36,38,45)(34,40,41,37)(35,42,44,48)(39,46,47,43);(1,3,6,12)(2,8,7,11)(4,15,14,10)(5,13,16,9)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,36)(34,37)(35,42)(38,45)(39,43)(40,41)(44,48)(46,47);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,8,16,11)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,5,6,16)(2,10,7,15)(3,13,12,9)(4,11,14,8)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,37)(34,36)(35,43)(38,40)(39,42)(41,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10078843);