Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8,6,3,10,9)(4,7,13,14,12,11,16,15)(17,18)(19,29)(20,25)(21,28)(22,27)(23,26)(24,31)(30,43)(32,42)(33,41)(34,40)(35,39)(36,38)(37,45)(44,48)(46,47);(1,3)(2,6)(4,11)(5,9)(7,12)(8,10)(13,15)(14,16)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,38)(27,29)(28,42)(33,44)(34,45)(35,46)(39,43)(40,48)(41,47);(1,4,10,16,6,12,5,13)(2,7,9,15,3,11,8,14)(17,20,23,33)(18,25,26,41)(19,30,32,44)(21,34,22,35)(24,37,38,47)(27,39,28,40)(29,43,42,48)(31,45,36,46);(2,3)(4,12)(8,9)(13,16)(18,26)(20,33)(24,38)(27,28)(29,42)(30,44)(34,35)(45,46);(1,5,6,10)(2,8,3,9)(4,13,12,16)(7,14,11,15)(17,21,23,22)(18,27,26,28)(19,31,32,36)(20,34,33,35)(24,29,38,42)(25,39,41,40)(30,45,44,46)(37,43,47,48)Polycyclic presentation:
2^9|a^2=g,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,d^a=dh,e^a=ehi,f^a=fh,d^b=di,f^b=fh,d^c=dh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252435);