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 | |
4366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,9,12,7)(5,8,14,10)(11,16,13,15)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,46)(45,47);(2,3)(7,9)(8,10)(15,16)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,35,38,37)(34,39,41,42)(36,43,44,45)(40,47,46,48);(2,3)(7,9)(8,10)(15,16)(19,26)(22,23)(27,32)(30,31)(36,44)(40,46)(43,45)(47,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,47)(41,46)(42,48);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,37,38,35)(34,39,41,42)(36,45,44,43)(40,47,46,48);(1,5)(2,8)(3,10)(4,11)(6,14)(7,15)(9,16)(12,13)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32)(33,34)(35,39)(36,46)(37,42)(38,41)(40,44)(43,48)(45,47)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=h,e^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, 8635572);