Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667760 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60570 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1643 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)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18,21,26)(19,29,32,24)(20,34,36,27)(22,25,23,28)(30,41,33,40)(31,39,47,45)(35,43,46,48)(37,44,38,42);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19,21,32)(18,24,26,29)(20,35,36,46)(22,33,23,30)(25,41,28,40)(27,43,34,48)(31,37,47,38)(39,42,45,44);(18,25)(19,30)(22,23)(26,28)(27,42)(31,46)(32,33)(34,44)(35,47)(37,38)(40,41)(43,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,26)(24,29)(25,28)(27,34)(39,45)(40,41)(42,44)(43,48);(3,11)(6,9)(10,16)(14,15)(17,20)(18,27)(19,31)(21,36)(22,37)(23,38)(24,39)(25,42)(26,34)(28,44)(29,45)(30,46)(32,47)(33,35)(40,48)(41,43)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579500);