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 | |
4080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,22,26)(19,25,29,23)(20,21,30,24)(27,28,32,31)(33,34)(35,41)(36,44)(37,43)(38,42)(39,40)(45,47)(46,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19,22,29)(18,23,26,25)(20,27,30,32)(21,31,24,28)(33,35)(34,41)(36,37)(38,45)(39,48)(40,46)(42,47)(43,44);(2,6)(7,10)(8,12)(14,16)(17,20,22,30)(18,21,26,24)(19,27,29,32)(23,31,25,28)(33,36,40,38)(34,42,39,44)(35,45,46,37)(41,43,48,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,27)(21,26)(22,30)(23,28)(25,31)(29,32)(33,37,40,45)(34,43,39,47)(35,36,46,38)(41,44,48,42);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,21,22,24)(18,20,26,30)(19,28,29,31)(23,32,25,27)(33,38,40,36)(34,44,39,42)(35,45,46,37)(41,43,48,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,19)(18,25)(20,27)(21,28)(22,29)(23,26)(24,31)(30,32)(33,39,40,34)(35,41,46,48)(36,44,38,42)(37,47,45,43)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8663171);