Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6358021 |
| 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 | |
1800 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)(3,13)(4,15)(5,10)(6,11)(7,12)(8,14)(9,16)(17,18,23,27)(19,28,32,43)(20,25,33,39)(21,26,22,36)(24,30,38,31)(29,42,44,48)(34,40,35,41)(37,46,47,45);(1,3,5,14,7,15,6,9)(2,8,11,13,12,16,10,4)(17,19,22,31,23,32,21,30)(18,24,26,28,27,38,36,43)(20,29,34,45,33,44,35,46)(25,37,41,42,39,47,40,48);(1,4,6,16,7,13,5,8)(2,9,10,15,12,14,11,3)(20,33)(25,39)(29,44)(34,35)(37,47)(40,41)(42,48)(45,46);(1,2)(3,8)(4,9)(5,11)(6,10)(7,12)(13,14)(15,16)(17,20)(18,25)(19,29)(21,35)(22,34)(23,33)(24,37)(26,41)(27,39)(28,42)(30,46)(31,45)(32,44)(36,40)(38,47)(43,48);(1,5,7,6)(2,10,12,11)(3,9,15,14)(4,8,13,16)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,34,33,35)(24,28,38,43)(25,40,39,41)(29,45,44,46)(37,48,47,42)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=ei,d^c=dhi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265250);