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 |
| prime radical | #47654 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4190 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)(3,12)(4,14)(5,11)(6,8)(7,9)(10,15)(13,16)(17,18,23,25)(19,24,28,32)(20,21,26,31)(22,30,29,27)(33,34,38,41)(35,37,43,39)(36,40,44,47)(42,45,48,46);(1,3,7,6)(2,8,9,12)(4,13,15,11)(5,10,16,14)(17,19,23,28)(18,24,25,32)(20,29,26,22)(21,27,31,30)(36,44)(40,47)(42,48)(45,46);(2,9)(4,15)(8,12)(11,13)(18,25)(19,28)(20,26)(27,30)(33,35,38,43)(34,39,41,37)(36,42,44,48)(40,46,47,45);(1,4)(2,10)(3,13)(5,12)(6,11)(7,15)(8,16)(9,14)(17,20)(18,21)(19,29)(22,28)(23,26)(24,27)(25,31)(30,32)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47);(1,5,7,16)(2,11,9,13)(3,14,6,10)(4,8,15,12)(17,21)(18,26)(19,30)(20,25)(22,32)(23,31)(24,29)(27,28)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,6,7,3)(2,12,9,8)(4,11,15,13)(5,14,16,10)(17,22,23,29)(18,27,25,30)(19,20,28,26)(21,24,31,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=ghi,e^2=g,f^2=gi,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, 10103312);