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 | |
4369 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)(3,11,13,8)(4,14,15,9)(5,10,16,12)(17,18,22,26)(19,21,25,23)(20,24,30,32)(27,29,28,31)(36,44)(40,45)(42,48)(46,47);(2,7)(4,15)(8,11)(10,12)(17,19,22,25)(18,21,26,23)(20,28,30,27)(24,31,32,29)(33,34)(35,39)(36,45)(37,41)(38,43)(40,44)(42,46)(47,48);(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,48)(45,47);(1,3,6,13)(2,8,7,11)(4,12,15,10)(5,14,16,9)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,30)(31,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,4,6,15)(2,9,7,14)(3,12,13,10)(5,11,16,8)(17,20,22,30)(18,24,26,32)(19,27,25,28)(21,29,23,31)(33,36)(34,40)(35,42)(37,44)(38,46)(39,47)(41,45)(43,48);(1,5)(2,10)(3,9)(4,11)(6,16)(7,12)(8,15)(13,14)(17,21)(18,25)(19,26)(20,29)(22,23)(24,28)(27,32)(30,31)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8379616);