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 | |
4089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(17,18,22,24)(19,23,26,20)(21,25,30,31)(27,29,32,28)(33,34,39,42)(35,40,45,36)(37,38,41,47)(43,44,48,46);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,19)(18,23)(20,24)(21,27)(22,26)(25,29)(28,31)(30,32)(33,35,39,45)(34,40,42,36)(37,43,41,48)(38,44,47,46);(4,11)(7,9)(12,16)(14,15)(18,24)(19,26)(25,31)(27,32)(33,36,39,40)(34,35,42,45)(37,46,41,44)(38,43,47,48);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,20)(18,19)(21,29)(22,23)(24,26)(25,32)(27,31)(28,30)(33,37)(34,38)(35,43)(36,46)(39,41)(40,44)(42,47)(45,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,31)(30,32)(33,36)(34,35)(37,46)(38,43)(39,40)(41,44)(42,45)(47,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,30)(18,25,24,31)(19,27,26,32)(20,28,23,29)(33,38)(34,41)(35,44)(36,43)(37,42)(39,47)(40,48)(45,46)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8440522);