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 | |
4174 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)(3,8,11,14)(4,13,15,9)(5,10,16,12)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,34)(35,37)(36,40)(38,41)(39,43)(42,45)(44,48)(46,47);(2,7)(3,11)(9,13)(10,12)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,46)(45,48);(2,7)(3,11)(9,13)(10,12)(19,26)(22,23)(27,32)(30,31)(36,44)(40,48)(42,46)(45,47);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,16)(10,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,36)(34,40)(35,42)(37,45)(38,44)(39,47)(41,48)(43,46);(1,4,6,15)(2,9,7,13)(3,12,11,10)(5,8,16,14)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,37,38,39)(34,35,41,43)(36,45,44,47)(40,42,48,46);(1,5,6,16)(2,10,7,12)(3,9,11,13)(4,14,15,8)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32)(33,35)(34,37)(36,46)(38,43)(39,41)(40,47)(42,44)(45,48)Polycyclic presentation:
2^9|a^2=h,e^2=g,f^2=h,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, 8635909);