Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500489 |
| 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 | C22×C4 | C25 |
| prime radical | #60755 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(18,23)(19,27)(21,33)(25,35)(26,36)(28,41)(29,30)(32,44)(34,38)(40,46)(42,43)(45,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,18,21,23,22,26,33,36)(19,24,29,34,30,39,27,38)(20,25,32,35,31,40,44,46)(28,37,43,45,42,47,41,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,34)(25,37)(26,39)(27,33)(31,42)(32,43)(35,45)(36,38)(40,47)(41,44)(46,48);(4,12)(7,16)(9,11)(13,15)(17,20)(18,25)(19,28)(21,32)(22,31)(23,35)(24,37)(26,40)(27,41)(29,43)(30,42)(33,44)(34,45)(36,46)(38,48)(39,47);(3,10)(6,14)(9,11)(13,15)(20,31)(25,40)(28,42)(32,44)(35,46)(37,47)(41,43)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6416108);