Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357716 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,9)(4,10)(6,12)(8,13)(11,15)(14,16)(17,18,22,26)(19,27,32,44)(20,24,33,40)(21,35,28,25)(23,36,39,31)(29,42,30,43)(34,48,45,41)(37,47,38,46);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,19,28,36,22,32,21,31)(18,23,35,44,26,39,25,27)(20,30,45,46,33,29,34,47)(24,38,48,42,40,37,41,43);(1,3,10,15,5,11,4,9)(2,6,13,16,7,14,8,12)(17,20)(18,24)(19,29)(21,34)(22,33)(23,37)(25,41)(26,40)(27,42)(28,45)(30,32)(31,46)(35,48)(36,47)(38,39)(43,44);(2,7)(6,14)(8,13)(12,16)(17,20)(18,24)(19,30)(21,34)(22,33)(23,38)(25,41)(26,40)(27,43)(28,45)(29,32)(31,47)(35,48)(36,46)(37,39)(42,44);(20,33)(24,40)(29,30)(34,45)(37,38)(41,48)(42,43)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=gi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,e^c=eh,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264931);