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 | #38676 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,10,4)(5,13,15,8)(11,12,16,14)(17,18)(19,24)(20,22)(21,26)(23,25)(27,32)(28,29)(30,31)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(2,7)(4,9)(8,13)(12,14)(17,19)(18,24)(20,28)(21,25)(22,29)(23,26)(27,31)(30,32)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(2,7)(3,10)(8,13)(11,16)(18,25)(19,26)(20,27)(29,30)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,14)(15,16)(17,20,23,27)(18,22,25,32)(19,28,26,31)(21,30,24,29)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,4)(2,3)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(17,21,23,24)(18,26,25,19)(20,30,27,29)(22,31,32,28)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,5,6,15)(2,8,7,13)(3,11,10,16)(4,12,9,14)(17,22)(18,27)(19,29)(20,25)(21,28)(23,32)(24,31)(26,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,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, 9619411);