Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806827 |
| 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 | #61262 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1569 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)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,24)(20,21)(22,37)(23,27)(25,39)(26,36)(28,32)(29,33)(30,44)(31,41)(34,48)(35,43)(38,47)(40,46)(42,45);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,24)(20,33)(21,29)(22,38)(23,32)(25,42)(26,41)(27,28)(30,35)(31,36)(34,40)(37,47)(39,45)(43,44)(46,48);(17,20,21,27,23,26,36,18)(19,28,31,33,32,24,29,41)(22,34,35,37,39,43,48,25)(30,44,38,47,46,40,45,42);(2,7)(6,9)(8,15)(14,16)(17,21,23,36)(18,20,27,26)(19,29,32,31)(22,35,39,48)(24,33,28,41)(25,34,37,43)(30,45,46,38)(40,47,44,42);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22,21,35,23,39,36,48)(18,25,20,34,27,37,26,43)(19,30,31,38,32,46,29,45)(24,40,41,42,28,44,33,47)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=i,e^2=fh,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=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716870);