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 | |
4089 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)(3,9,4,12)(5,15,16,6)(10,14,13,11)(18,20)(23,26)(24,28)(30,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47);(2,8)(5,16)(9,12)(10,13)(19,25)(23,26)(27,31)(30,32)(33,34)(35,43)(36,47)(37,48)(38,42)(39,46)(40,45)(41,44);(1,3,7,4)(2,9,8,12)(5,10,16,13)(6,11,15,14)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,41)(42,43)(47,48);(1,4,7,3)(2,9,8,12)(5,13,16,10)(6,11,15,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,41,43,48);(1,5)(2,6)(3,10)(4,13)(7,16)(8,15)(9,11)(12,14)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,40,43,47);(1,6)(2,5)(3,11)(4,14)(7,15)(8,16)(9,10)(12,13)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,34)(35,39)(36,47)(37,41)(38,42)(40,45)(43,46)(44,48)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,d^2=gi,e^2=h,f^2=hi,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, 8411956);