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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,16,14)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,30)(29,31)(33,34,37,40)(35,41,42,38)(36,39,46,45)(43,48,44,47);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,31,30,32)(35,42)(38,41)(43,44)(47,48);(3,11)(7,8)(12,16)(13,14)(20,28)(24,30)(27,29)(31,32)(33,35,37,42)(34,38,40,41)(36,43,46,44)(39,47,45,48);(1,3)(2,8)(4,12)(5,14)(6,11)(7,10)(9,16)(13,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,30)(26,32)(33,35)(34,38)(36,44)(37,42)(39,48)(40,41)(43,46)(45,47);(1,4)(2,5)(3,12)(6,9)(7,13)(8,14)(10,15)(11,16)(17,21,22,19)(18,23,25,26)(20,29,28,27)(24,31,30,32)(33,34)(35,38)(36,45)(37,40)(39,46)(41,42)(43,48)(44,47);(1,5)(2,9)(3,13)(4,10)(6,15)(7,16)(8,12)(11,14)(17,18)(19,23)(20,30)(21,26)(22,25)(24,28)(27,32)(29,31)(33,36)(34,39)(35,43)(37,46)(38,47)(40,45)(41,48)(42,44)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,e^2=g,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, 8637964);