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 | |
3985 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,8,10,5)(4,9,13,14)(11,15,16,12)(17,18)(19,23)(20,27)(21,24)(22,25)(26,31)(28,32)(29,30)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,45)(47,48);(1,3,7,10)(2,8,6,5)(4,11,13,16)(9,15,14,12)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28)(34,39)(35,42)(41,44)(43,48);(3,10)(5,8)(11,16)(12,15)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4)(2,9)(3,11)(5,12)(6,14)(7,13)(8,15)(10,16)(17,18)(19,20)(21,29)(22,25)(23,27)(24,30)(26,31)(28,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5)(2,3)(4,12)(6,10)(7,8)(9,11)(13,15)(14,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31);(1,6,7,2)(3,5,10,8)(4,9,13,14)(11,15,16,12)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=g,f^2=hi,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, 8458660);