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 | |
4150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34,40,44)(35,41,46,47)(36,38,37,42)(39,45,48,43);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,18,22,26)(19,23,20,24)(21,25,29,32)(27,30,28,31)(33,35)(34,41)(36,43)(37,45)(38,39)(40,46)(42,48)(44,47);(2,6)(4,12)(7,10)(11,15)(17,19,22,20)(18,23,26,24)(21,28,29,27)(25,31,32,30)(33,36,40,37)(34,42,44,38)(35,45,46,43)(39,47,48,41);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30)(33,37,40,36)(34,38,44,42)(35,45,46,43)(39,47,48,41);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18)(19,24)(20,23)(21,25)(22,26)(27,31)(28,30)(29,32)(33,38)(34,37)(35,39)(36,44)(40,42)(41,45)(43,47)(46,48);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,39)(34,43)(35,38)(36,41)(37,47)(40,48)(42,46)(44,45)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8946476);