Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| 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 | #61433 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2965 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,9,6,3,7,11)(4,8,13,15,12,10,14,16)(17,18)(19,27)(20,31)(21,34)(22,24)(23,29)(25,32)(26,36)(28,42)(30,40)(33,39)(35,37)(38,48)(41,43)(44,45)(46,47);(17,19,22,30)(18,23,24,37)(20,28,32,43)(21,35,36,29)(25,38,31,45)(26,40,34,27)(33,48,47,44)(39,42,46,41);(1,3,5,11,6,2,7,9)(4,10,13,16,12,8,14,15)(18,24)(20,32)(23,37)(26,34)(27,40)(28,43)(33,47)(44,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,20)(18,25)(19,28)(21,33)(22,32)(23,38)(24,31)(26,39)(27,41)(29,44)(30,43)(34,46)(35,48)(36,47)(37,45)(40,42);(4,12)(8,10)(13,14)(15,16)(20,32)(25,31)(28,43)(33,47)(38,45)(39,46)(41,42)(44,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=dh,f^a=fh,f^b=fh,d^c=dh,e^d=ehi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72321740, #72321751, #72321918, … |
| maximal normal subgroup of | #72321740, #72321751, #72321918, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253103);