Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1734 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)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18,21,27,23,28,22,26)(19,29,31,24,33,46,32,40)(20,25,36,44,38,45,37,43)(30,41,47,39,35,48,34,42);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19,23,33)(18,24,28,40)(20,34,38,47)(21,32,22,31)(25,41,45,48)(26,46,27,29)(30,36,35,37)(39,44,42,43);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,19)(18,24)(20,35)(21,31)(22,32)(23,33)(25,42)(26,29)(27,46)(28,40)(30,38)(34,36)(37,47)(39,45)(41,44)(43,48);(4,12)(7,16)(9,11)(13,15)(17,20,22,37,23,38,21,36)(18,25,26,43,28,45,27,44)(19,30,32,34,33,35,31,47)(24,39,29,41,40,42,46,48);(2,8)(3,10)(7,16)(9,11)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,38,37)(24,29,40,46)(25,43,45,44)(30,47,35,34)(39,41,42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=i,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6941826);