Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| 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 | #61447 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2965 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,3,6,8,13,10)(4,7,12,9,11,14,16,15)(17,18,23,28)(19,29,32,42)(20,25,33,39)(21,27,22,26)(24,36,38,31)(30,43,44,48)(34,41,35,40)(37,45,47,46);(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,29)(27,42)(28,38)(33,44)(34,45)(35,46)(39,47)(40,48)(41,43);(1,3,5,8,6,10,13,2)(4,9,12,14,11,15,16,7)(20,33)(25,39)(30,44)(34,35)(37,47)(40,41)(43,48)(45,46);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20,23,33)(18,25,28,39)(19,30,32,44)(21,35,22,34)(24,37,38,47)(26,41,27,40)(29,43,42,48)(31,46,36,45);(4,11)(7,14)(9,15)(12,16)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,34,33,35)(24,29,38,42)(25,40,39,41)(30,45,44,46)(37,48,47,43)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ci,e^a=eh,f^a=fh,f^b=fh,d^c=dh,e^d=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253504);