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 | |
4222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,11,14)(4,13,15,9)(5,10,16,12)(17,18,21,22)(19,25,27,23)(20,24,30,28)(26,32,29,31)(33,34)(35,40)(36,39)(37,41)(38,42)(43,45)(44,47)(46,48);(2,7)(3,11)(9,13)(10,12)(17,18,21,22)(19,23,27,25)(20,24,30,28)(26,31,29,32)(33,35)(34,37)(36,43)(38,41)(39,46)(40,42)(44,48)(45,47);(2,7)(3,11)(9,13)(10,12)(18,22)(23,25)(24,28)(31,32)(36,44)(39,47)(43,48)(45,46);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,16)(10,15)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,36)(34,39)(35,43)(37,46)(38,44)(40,45)(41,48)(42,47);(1,4,6,15)(2,9,7,13)(3,12,11,10)(5,8,16,14)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31)(33,35)(34,40)(36,43)(37,42)(38,41)(39,45)(44,48)(46,47);(1,5,6,16)(2,10,7,12)(3,9,11,13)(4,14,15,8)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)(33,37,38,40)(34,41,42,35)(36,45,44,46)(39,43,47,48)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=h,f^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, 8641896);