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 | |
4024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,8)(3,13)(4,12)(11,15)(17,18)(19,28)(20,25)(21,29)(22,30)(23,26)(24,27)(31,32)(33,34,38,40)(35,41,42,36)(37,39,46,48)(43,45,47,44);(1,2,7,8)(3,9,13,16)(4,6,12,10)(5,11,14,15)(17,19,24,26)(18,23,27,28)(20,29,31,22)(21,25,30,32)(35,42)(36,41)(43,47)(44,45);(1,3,7,13)(2,9,8,16)(4,14,12,5)(6,15,10,11)(17,20,24,31)(18,25,27,32)(19,29,26,22)(21,23,30,28)(33,35)(34,36)(37,43)(38,42)(39,44)(40,41)(45,48)(46,47);(1,4)(2,10)(3,14)(5,13)(6,8)(7,12)(9,11)(15,16)(17,21)(18,22)(19,25)(20,23)(24,30)(26,32)(27,29)(28,31)(33,36)(34,35)(37,45)(38,41)(39,47)(40,42)(43,48)(44,46);(1,5,7,14)(2,11,8,15)(3,4,13,12)(6,16,10,9)(17,22)(18,21)(19,20)(23,25)(24,29)(26,31)(27,30)(28,32);(1,6,7,10)(2,12,8,4)(3,15,13,11)(5,16,14,9)(17,23)(18,26)(19,27)(20,30)(21,31)(22,25)(24,28)(29,32)(33,37)(34,39)(35,43)(36,44)(38,46)(40,48)(41,45)(42,47)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8367896);