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 | |
4269 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)(3,10)(4,7)(5,8)(6,12)(9,14)(11,15)(13,16)(17,18,23,20)(19,21,24,26)(22,25,31,28)(27,29,30,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(1,3,7,5)(2,8,4,10)(6,13,14,11)(9,16,12,15)(17,19,23,24)(18,21,20,26)(22,30,31,27)(25,32,28,29)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(1,2,7,4)(3,8,5,10)(6,9,14,12)(11,15,13,16)(17,18,23,20)(19,26,24,21)(22,25,31,28)(27,32,30,29)(33,34,37,40)(35,38,43,41)(36,39,45,44)(42,46,48,47);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,20,23,18)(19,21,24,26)(22,28,31,25)(27,29,30,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,21)(18,24)(19,20)(22,29)(23,26)(25,30)(27,28)(31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,30)(26,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10275818);