Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470067 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(19,27)(21,32)(23,35)(25,33)(28,41)(29,34)(31,43)(36,46)(37,40)(39,44)(42,45)(47,48);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,18,22,26)(19,23,29,37)(20,24,30,38)(21,33,32,25)(27,40,34,35)(28,36,42,47)(31,44,43,39)(41,48,45,46);(3,9)(6,13)(10,11)(14,15)(17,19)(18,23)(20,28)(21,34)(22,29)(24,36)(25,40)(26,37)(27,32)(30,42)(31,45)(33,35)(38,47)(39,48)(41,43)(44,46);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,20,22,30)(18,24,26,38)(19,28,29,42)(21,31,32,43)(23,36,37,47)(25,39,33,44)(27,41,34,45)(35,46,40,48);(2,8)(6,13)(7,16)(14,15)(20,30)(24,38)(28,42)(31,43)(36,47)(39,44)(41,45)(46,48)Polycyclic presentation:
2^9|b^2=i,d^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^b=gi,f^c=fi,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, 6388685);