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 | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4090 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18,23,25)(19,24,28,22)(20,26,30,31)(21,27,29,32)(33,34,39,44)(35,43,38,40)(36,46,47,41)(37,42,48,45);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19,23,28)(18,24,25,22)(20,29,30,21)(26,32,31,27)(33,35,39,38)(34,40,44,43)(36,45,47,42)(37,46,48,41);(2,4)(3,10)(9,12)(11,16)(18,25)(19,28)(20,30)(27,32)(33,36,39,47)(34,41,44,46)(35,45,38,42)(37,43,48,40);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20)(18,26)(19,29)(21,28)(22,27)(23,30)(24,32)(25,31)(33,36)(34,41)(35,45)(37,40)(38,42)(39,47)(43,48)(44,46);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,21,23,29)(18,27,25,32)(19,30,28,20)(22,26,24,31)(33,37)(34,42)(35,41)(36,43)(38,46)(39,48)(40,47)(44,45);(1,6,7,14)(2,9,4,12)(3,11,10,16)(5,13,8,15)(17,22,23,24)(18,28,25,19)(20,27,30,32)(21,31,29,26)(33,38,39,35)(34,43,44,40)(36,42,47,45)(37,41,48,46)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=hi,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10050801);