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 |
| prime radical | #54949 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4094 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,11,7)(5,8,14,12)(10,16,13,15)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,34)(35,44)(36,41)(37,42)(38,40)(39,43)(45,46)(47,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35,39,38)(34,40,43,44)(36,45,42,47)(37,46,41,48);(2,4)(7,9)(8,12)(15,16)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31)(33,36,39,42)(34,41,43,37)(35,45,38,47)(40,48,44,46);(1,4,6,2)(3,9,11,7)(5,8,14,12)(10,15,13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,36)(34,41)(35,45)(37,43)(38,47)(39,42)(40,48)(44,46);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32)(33,37,39,41)(34,42,43,36)(35,46,38,48)(40,47,44,45);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,38,39,35)(34,40,43,44)(36,47,42,45)(37,46,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9091109);