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 | |
4095 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,11,7)(4,8,14,12)(10,16,13,15)(17,18)(19,23)(20,25)(21,26)(22,27)(24,28)(29,31)(30,32)(33,34,39,41)(35,44,38,40)(36,42,46,48)(37,45,47,43);(1,2,5,6)(3,7,11,9)(4,8,14,12)(10,15,13,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,31)(26,32)(33,35,39,38)(34,40,41,44)(36,43,46,45)(37,42,47,48);(2,6)(7,9)(8,12)(15,16)(19,28)(23,24)(29,32)(30,31)(34,41)(36,46)(40,44)(43,45);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,19)(18,24)(20,29)(21,31)(22,28)(23,27)(25,30)(26,32)(33,36,39,46)(34,42,41,48)(35,45,38,43)(37,44,47,40);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20)(18,25)(19,29)(21,27)(22,26)(23,31)(24,30)(28,32)(33,37)(34,43)(35,42)(36,44)(38,48)(39,47)(40,46)(41,45);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,21,22,25)(18,26,27,20)(19,30,28,31)(23,32,24,29)(33,38,39,35)(34,40,41,44)(36,45,46,43)(37,42,47,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=g,e^2=gh,f^2=gi,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, 8653737);