Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356901 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62136 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,12)(4,10)(5,11)(6,8)(7,15)(9,13)(14,16)(17,18,22,26)(19,27,29,42)(20,24,33,40)(21,35,28,25)(23,36,37,31)(30,43,32,46)(34,48,44,41)(38,47,39,45);(2,8)(9,16)(10,12)(11,15)(17,19,28,36,22,29,21,31)(18,23,35,42,26,37,25,27)(20,32,44,45,33,30,34,47)(24,39,48,43,40,38,41,46);(1,3,6,4)(2,9,8,16)(5,13,7,14)(10,15,12,11)(19,29)(23,37)(27,42)(30,32)(31,36)(38,39)(43,46)(45,47);(1,4,6,3)(2,10,8,12)(5,13,7,14)(9,15,16,11)(17,20)(18,24)(19,30)(21,34)(22,33)(23,38)(25,41)(26,40)(27,43)(28,44)(29,32)(31,45)(35,48)(36,47)(37,39)(42,46);(1,5,6,7)(2,11,8,15)(3,13,4,14)(9,10,16,12)(20,33)(24,40)(30,32)(34,44)(38,39)(41,48)(43,46)(45,47)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=i,d^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264262);