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 | |
4130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18,22,27)(19,23,30,24)(20,25,26,21)(28,29,31,32)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19)(18,23)(20,31)(21,29)(22,30)(24,27)(25,32)(26,28)(33,35)(34,36)(37,45)(38,39)(40,47)(41,42)(43,46)(44,48);(2,6)(7,9)(8,12)(14,15)(17,19,22,30)(18,24,27,23)(20,28,26,31)(21,29,25,32)(33,36,38,42)(34,39,41,35)(37,44,46,47)(40,45,48,43);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,27)(25,32)(26,31)(33,36)(34,35)(37,44)(38,42)(39,41)(40,43)(45,48)(46,47);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20)(18,25)(19,28)(21,27)(22,26)(23,29)(24,32)(30,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,45)(41,48)(42,47);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21,22,25)(18,26,27,20)(19,29,30,32)(23,28,24,31)(33,35)(34,36)(37,45)(38,39)(40,47)(41,42)(43,46)(44,48)Polycyclic presentation:
2^9|a^2=h,c^2=gi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9627711);