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)(3,4,11,8)(5,9,14,15)(10,12,13,16)(20,28)(24,29)(26,32)(30,31)(33,34,40,43)(35,36,44,46)(37,41,45,48)(38,39,42,47);(2,7)(4,8)(9,15)(12,16)(17,18)(19,23)(20,29)(21,25)(22,27)(24,28)(26,30)(31,32)(33,35,40,44)(34,36,43,46)(37,47,45,39)(38,48,42,41);(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31)(33,36)(34,35)(37,42)(38,45)(39,41)(40,46)(43,44)(47,48);(1,3)(2,8)(4,7)(5,13)(6,11)(9,12)(10,14)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,37,40,45)(34,41,43,48)(35,39,44,47)(36,42,46,38);(1,4,6,8)(2,3,7,11)(5,12,14,16)(9,10,15,13)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,29)(27,32)(33,38,40,42)(34,39,43,47)(35,41,44,48)(36,45,46,37);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,39)(34,42)(35,37)(36,41)(38,43)(40,47)(44,45)(46,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=h,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, 8380096);