Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6358057 |
| 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 |
| prime radical | #62541 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,9)(4,11)(6,12)(8,14)(10,15)(13,16)(17,18,23,27)(19,28,29,42)(20,25,33,39)(21,26,22,36)(24,31,37,32)(30,43,44,48)(34,40,35,41)(38,46,47,45);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,19,21,31,23,29,22,32)(18,24,36,42,27,37,26,28)(20,30,35,46,33,44,34,45)(25,38,40,48,39,47,41,43);(1,3,4,9,5,10,11,15)(2,6,8,12,7,13,14,16)(19,29)(20,33)(24,37)(25,39)(28,42)(31,32)(34,35)(40,41);(3,10)(6,13)(9,15)(12,16)(17,20)(18,25)(19,30)(21,35)(22,34)(23,33)(24,38)(26,41)(27,39)(28,43)(29,44)(31,46)(32,45)(36,40)(37,47)(42,48);(2,7)(6,13)(8,14)(12,16)(17,21,23,22)(18,26,27,36)(19,31,29,32)(20,34,33,35)(24,28,37,42)(25,40,39,41)(30,45,44,46)(38,48,47,43)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,e^b=ei,d^c=dhi,e^d=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265270);