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 | #49590 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4253 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)(3,7,10,8)(5,9,13,12)(11,14,16,15)(17,18)(19,24)(20,23)(21,25)(22,27)(26,32)(28,31)(29,30)(35,42)(37,45)(39,40)(41,44);(1,3)(2,7)(4,8)(5,11)(6,10)(9,14)(12,15)(13,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,34)(35,40)(36,38)(37,41)(39,42)(43,47)(44,45)(46,48);(3,10)(7,8)(11,16)(14,15)(20,28)(23,31)(26,29)(30,32)(33,35,38,42)(34,39,36,40)(37,43,45,48)(41,46,44,47);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,14)(13,16)(17,20,21,28)(18,23,25,31)(19,26,27,29)(22,30,24,32)(33,35)(34,40)(36,39)(37,43)(38,42)(41,47)(44,46)(45,48);(1,4,6,2)(3,8,10,7)(5,9,13,12)(11,14,16,15)(17,19)(18,24)(20,29)(21,27)(22,25)(23,30)(26,28)(31,32)(33,36,38,34)(35,40,42,39)(37,41,45,44)(43,46,48,47);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,31)(29,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=g,e^2=i,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, 9110970);