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 | |
4145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,9,4,14)(5,6,11,10)(12,13,15,16)(20,28)(24,29)(26,32)(30,31)(33,34)(35,37)(36,45)(38,41)(39,42)(40,47)(43,48)(44,46);(2,8)(5,11)(9,14)(12,15)(17,18)(19,23)(20,29)(21,25)(22,27)(24,28)(26,30)(31,32)(33,35,39,44)(34,37,42,46)(36,41,47,43)(38,40,48,45);(1,3,7,4)(2,9,8,14)(5,12,11,15)(6,13,10,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31)(33,36,39,47)(34,40,42,45)(35,43,44,41)(37,38,46,48);(1,4,7,3)(2,9,8,14)(5,15,11,12)(6,13,10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,37)(34,35)(36,38)(39,46)(40,43)(41,45)(42,44)(47,48);(1,5,7,11)(2,10,8,6)(3,12,4,15)(9,16,14,13)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,29)(27,32)(33,35)(34,37)(36,43)(38,40)(39,44)(41,47)(42,46)(45,48);(1,6,7,10)(2,11,8,5)(3,13,4,16)(9,15,14,12)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,38)(34,41)(35,40)(36,46)(37,47)(39,48)(42,43)(44,45)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=g,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, 8378832);