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 | |
4273 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18,22,24)(19,27,29,23)(20,25,30,28)(21,31,32,26)(33,34,39,42)(35,40,36,38)(37,41,46,48)(43,47,44,45);(2,3)(5,13)(7,9)(12,16)(17,19,22,29)(18,23,24,27)(20,26,30,31)(21,25,32,28)(35,36)(38,40)(43,44)(45,47);(4,11)(7,9)(12,16)(14,15)(18,24)(20,30)(23,27)(26,31)(33,35,39,36)(34,40,42,38)(37,44,46,43)(41,45,48,47);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,23)(20,26)(21,28)(22,29)(24,27)(25,32)(30,31)(33,36,39,35)(34,38,42,40)(37,43,46,44)(41,47,48,45);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20)(18,25)(19,26)(21,27)(22,30)(23,32)(24,28)(29,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,32)(18,26,24,31)(19,28,29,25)(20,27,30,23)(33,38)(34,35)(36,42)(37,45)(39,40)(41,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10270937);