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 | |
4148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,10,9,6)(5,8,13,12)(11,16,15,14)(17,18)(19,23)(20,27)(21,29)(22,25)(24,31)(26,28)(30,32)(33,34)(35,37)(36,38)(39,41)(40,44)(42,46)(43,47)(45,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,16)(13,15)(17,19,22,27)(18,23,25,20)(21,26,31,32)(24,30,29,28)(33,35,39,44)(34,40,41,37)(36,42,45,47)(38,43,48,46);(2,4)(5,13)(6,10)(11,15)(17,20,22,23)(18,19,25,27)(21,30,31,28)(24,32,29,26)(33,36,39,45)(34,38,41,48)(35,42,44,47)(37,46,40,43);(1,4,7,2)(3,10,9,6)(5,8,13,12)(11,14,15,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,36)(34,38)(35,42)(37,46)(39,45)(40,43)(41,48)(44,47);(1,5,7,13)(2,8,4,12)(3,11,9,15)(6,14,10,16)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,32)(29,31)(33,37,39,40)(34,35,41,44)(36,43,45,46)(38,47,48,42);(1,6,7,10)(2,9,4,3)(5,14,13,16)(8,15,12,11)(17,21,22,31)(18,24,25,29)(19,26,27,32)(20,28,23,30)(33,38)(34,36)(35,43)(37,47)(39,48)(40,42)(41,45)(44,46)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gh,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8957570);