Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470058 |
| 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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2919 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)(17,18)(19,24)(20,32)(21,28)(22,27)(23,26)(25,34)(29,45)(30,40)(31,43)(33,48)(35,44)(36,39)(37,42)(38,47)(41,46);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19)(18,24)(20,33)(21,31)(22,36)(23,30)(25,41)(26,40)(27,39)(28,43)(29,35)(32,45)(34,47)(37,46)(38,42)(44,48);(3,9)(6,13)(10,11)(14,15)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,38)(26,42)(27,32)(28,44)(30,33)(31,46)(36,47)(39,48)(40,41)(43,45);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(18,26)(19,30)(25,42)(27,28)(29,33)(31,36)(32,44)(46,47);(2,8)(6,13)(7,16)(14,15)(17,21,23,22)(18,27,26,28)(19,31,30,36)(20,34,35,37)(24,39,40,43)(25,32,42,44)(29,46,33,47)(38,48,41,45)Polycyclic presentation:
2^9|d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388680);