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 | |
4106 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)(3,10)(4,5)(6,7)(8,11)(9,15)(12,13)(14,16)(17,18,22,24)(19,23,25,20)(21,29,31,26)(27,28,30,32)(33,34)(35,39)(36,40)(37,46)(38,42)(41,47)(43,44)(45,48);(2,7)(5,9)(8,10)(13,16)(17,19)(18,23)(20,24)(21,30)(22,25)(26,28)(27,31)(29,32)(33,35)(34,39)(36,37)(38,43)(40,46)(41,48)(42,44)(45,47);(3,11)(8,10)(12,14)(13,16)(18,24)(19,25)(26,29)(27,30)(35,43)(36,45)(39,44)(40,48);(1,3,6,11)(2,8,7,10)(4,14,15,12)(5,13,9,16)(17,19)(18,23)(20,24)(21,27)(22,25)(26,32)(28,29)(30,31)(33,36,38,45)(34,40,42,48)(35,37,43,47)(39,46,44,41);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,20,22,23)(18,25,24,19)(21,28,31,32)(26,30,29,27)(33,37)(34,41)(35,36)(38,47)(39,48)(40,44)(42,46)(43,45);(1,5,6,9)(2,4,7,15)(3,13,11,16)(8,14,10,12)(17,21,22,31)(18,26,24,29)(19,27,25,30)(20,28,23,32)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=h,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746261);