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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3811 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,4,11,7)(5,8,14,13)(10,12,16,15)(17,18,23,26)(19,24,22,20)(21,29,31,25)(27,28,30,32)(33,34,39,37)(35,40,46,45)(36,42,47,41)(38,48,44,43);(1,3,6,11)(2,7,9,4)(5,10,14,16)(8,15,13,12)(17,19,23,22)(18,24,26,20)(21,30,31,27)(25,28,29,32)(33,35,39,46)(34,40,37,45)(36,44,47,38)(41,48,42,43);(17,20)(18,22)(19,26)(21,28)(23,24)(25,30)(27,29)(31,32)(33,36)(34,41)(35,44)(37,42)(38,46)(39,47)(40,48)(43,45);(1,2)(3,7)(4,11)(5,13)(6,9)(8,14)(10,12)(15,16)(17,20)(18,19)(21,28)(22,26)(23,24)(25,27)(29,30)(31,32)(33,37,39,34)(35,45,46,40)(36,42,47,41)(38,48,44,43);(1,4,6,7)(2,3,9,11)(5,12,14,15)(8,10,13,16)(17,21,23,31)(18,25,26,29)(19,27,22,30)(20,28,24,32)(33,36)(34,42)(35,44)(37,41)(38,46)(39,47)(40,43)(45,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,13)(11,16)(17,22,23,19)(18,24,26,20)(21,30,31,27)(25,32,29,28)(33,38,39,44)(34,43,37,48)(35,36,46,47)(40,41,45,42)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=hi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10349724);