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 | |
4009 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)(3,8,12,7)(4,13,14,5)(10,11,16,15)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,32)(30,31)(35,37)(36,40)(43,45)(44,46);(1,3,6,12)(2,7,9,8)(4,10,14,16)(5,11,13,15)(17,19,22,28)(18,23,26,27)(20,25,29,30)(21,24,31,32)(33,34,39,42)(35,36,37,40)(38,41,47,48)(43,44,45,46);(1,3,6,12)(2,7,9,8)(4,10,14,16)(5,11,13,15)(17,20,22,29)(18,24,26,32)(19,25,28,30)(21,23,31,27)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,46,48,44);(1,3)(2,8)(4,10)(5,15)(6,12)(7,9)(11,13)(14,16)(17,20)(18,24)(19,25)(21,27)(22,29)(23,31)(26,32)(28,30)(33,36,39,40)(34,35,42,37)(38,46,47,44)(41,45,48,43);(1,4,6,14)(2,5,9,13)(3,10,12,16)(7,11,8,15)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,28)(29,32)(33,37,39,35)(34,40,42,36)(38,43,47,45)(41,44,48,46);(1,5)(2,4)(3,11)(6,13)(7,10)(8,16)(9,14)(12,15)(17,21,22,31)(18,25,26,30)(19,24,28,32)(20,23,29,27)(33,38,39,47)(34,41,42,48)(35,43,37,45)(36,44,40,46)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=gi,d^2=i,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653180);