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 | #53870 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3956 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)(3,7)(4,10)(5,9)(6,8)(11,14)(12,13)(15,16)(17,18)(19,20)(21,31)(22,27)(23,25)(24,29)(26,28)(30,32)(33,34,38,42)(35,37,41,39)(36,40,44,47)(43,45,48,46);(1,3,6,10)(2,7,8,4)(5,11,15,13)(9,14,16,12)(17,19,23,29)(18,24,25,20)(21,22,28,32)(26,27,31,30)(36,44)(40,47)(43,48)(45,46);(2,8)(3,10)(9,16)(11,13)(18,25)(20,24)(26,31)(27,30)(33,35,38,41)(34,39,42,37)(36,43,44,48)(40,46,47,45);(1,3)(2,4)(5,13)(6,10)(7,8)(9,14)(11,15)(12,16)(17,20,23,24)(18,19,25,29)(21,30,28,27)(22,31,32,26)(33,36,38,44)(34,40,42,47)(35,43,41,48)(37,45,39,46);(1,4)(2,3)(5,14)(6,7)(8,10)(9,13)(11,16)(12,15)(17,21)(18,26)(19,22)(20,30)(23,28)(24,27)(25,31)(29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,6,15)(2,9,8,16)(3,11,10,13)(4,12,7,14)(17,22)(18,27)(19,28)(20,26)(21,29)(23,32)(24,31)(25,30)(33,37,38,39)(34,41,42,35)(36,45,44,46)(40,48,47,43)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653114);