Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10508 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | #60954 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,10,7,3,6,9)(4,12,14,8,15,13,16,11)(17,18)(19,29,32,24)(20,33)(21,28)(22,27)(23,26)(25,36)(30,46,47,41)(31,40,38,44)(34,45,48,39)(35,42)(37,43);(1,3)(2,7)(4,13)(5,9)(6,10)(8,16)(11,14)(12,15)(17,19)(18,24)(20,34)(21,31)(22,38)(23,32)(25,41)(26,29)(27,40)(28,44)(30,37)(33,45)(35,48)(36,47)(39,42)(43,46);(1,4)(2,8)(3,11)(5,14)(6,16)(7,15)(9,12)(10,13)(17,20)(18,25)(19,30)(21,37)(22,36)(23,35)(24,39)(26,43)(27,42)(28,33)(29,45)(31,48)(32,47)(34,38)(40,46)(41,44)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=dhi,e^c=eh,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 20433);