Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| 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 | #61441 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2922 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,3,5,7,13,9)(4,8,12,10,11,14,16,15)(17,18)(19,28)(20,32)(21,26)(22,36)(23,27)(24,30)(25,33)(29,43)(31,39)(34,47)(35,41)(37,40)(38,48)(42,44)(45,46);(2,7)(3,9)(8,14)(10,15)(17,19)(18,24)(20,29)(21,30)(22,37)(23,31)(25,38)(26,39)(27,40)(28,36)(32,46)(33,44)(34,45)(35,48)(41,43)(42,47);(1,3,6,7,5,9,13,2)(4,10,12,14,11,15,16,8)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,38)(26,41)(27,32)(28,42)(30,45)(31,44)(36,47)(37,48)(39,43)(40,46);(1,4)(2,8)(3,10)(5,11)(6,12)(7,14)(9,15)(13,16)(20,33)(25,32)(29,44)(34,35)(38,46)(41,47)(42,43)(45,48);(4,11)(8,14)(10,15)(12,16)(17,21,23,22)(18,26,27,36)(19,30,31,37)(20,34,33,35)(24,39,40,28)(25,41,32,47)(29,45,44,48)(38,43,46,42)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=chi,f^a=fh,c^b=ci,f^b=fh,d^c=dh,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253144);