Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6473174 |
| 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 | #60807 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1644 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)(3,6)(4,11)(5,7)(8,12)(9,13)(10,16)(14,15)(17,18,23,27)(19,28,33,41)(20,34,35,48)(21,25,37,36)(22,38,39,26)(24,32,42,31)(29,30,44,46)(40,47,45,43);(1,3)(2,6)(4,10)(5,9)(7,13)(8,14)(11,16)(12,15)(17,19)(18,24)(20,29)(21,30)(22,32)(23,33)(25,40)(26,41)(27,42)(28,38)(31,39)(34,47)(35,44)(36,45)(37,46)(43,48);(2,7)(3,9)(8,11)(10,15)(17,20,22,36,23,35,39,25)(18,21,26,34,27,37,38,48)(19,29,32,45,33,44,31,40)(24,30,41,47,42,46,28,43);(3,9)(6,13)(10,15)(14,16)(17,21)(18,25)(19,30)(20,26)(22,34)(23,37)(24,40)(27,36)(28,44)(29,41)(31,43)(32,47)(33,46)(35,38)(39,48)(42,45);(1,4,5,12)(2,8,7,11)(3,10,9,15)(6,14,13,16)(20,35)(21,37)(25,36)(29,44)(30,46)(34,48)(40,45)(43,47)Polycyclic presentation:
2^9|a^2=i,c^2=gh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405703);