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 | |
4081 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)(3,7)(4,11)(5,13)(6,8)(9,14)(10,16)(12,15)(17,18,22,26)(19,24,23,20)(21,29,31,25)(27,28,32,30)(33,34,39,44)(35,43,38,40)(36,37,42,41)(45,48,47,46);(1,3,6,4)(2,7,8,11)(5,10,14,12)(9,15,13,16)(17,19)(18,20)(21,27)(22,23)(24,26)(25,28)(29,30)(31,32)(33,35,39,38)(34,40,44,43)(36,47,42,45)(37,48,41,46);(2,8)(3,4)(5,14)(15,16)(17,20,22,24)(18,23,26,19)(21,30,31,28)(25,27,29,32)(33,36,39,42)(34,41,44,37)(35,45,38,47)(40,48,43,46);(1,4,6,3)(2,7,8,11)(5,12,14,10)(9,15,13,16)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,32)(30,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47);(1,2)(3,7)(4,11)(5,9)(6,8)(10,15)(12,16)(13,14)(17,19)(18,24)(20,26)(21,27)(22,23)(25,30)(28,29)(31,32)(33,37,39,41)(34,42,44,36)(35,46,38,48)(40,47,43,45);(1,5,6,14)(2,9,8,13)(3,10,4,12)(7,15,11,16)(17,21,22,31)(18,25,26,29)(19,27,23,32)(20,28,24,30)(33,38,39,35)(34,43,44,40)(36,47,42,45)(37,48,41,46)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=gh,e^2=ghi,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, 8961613);