Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61569 |
| 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 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1514 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)(17,18,23,25)(19,28,31,40)(20,32,34,26)(21,27,22,36)(24,37,38,30)(29,44,47,43)(33,42,35,41)(39,46,48,45);(17,19,23,31)(18,24,25,38)(20,29,34,47)(21,30,22,37)(26,39,32,48)(27,40,36,28)(33,45,35,46)(41,43,42,44);(1,3,5,8,6,10,13,2)(4,9,12,14,11,15,16,7)(18,25)(24,38)(26,32)(27,36)(28,40)(39,48)(41,42)(43,44);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20)(18,26)(19,29)(21,35)(22,33)(23,34)(24,39)(25,32)(27,42)(28,43)(30,46)(31,47)(36,41)(37,45)(38,48)(40,44);(4,11)(7,14)(9,15)(12,16)(17,21,23,22)(18,27,25,36)(19,30,31,37)(20,33,34,35)(24,40,38,28)(26,41,32,42)(29,45,47,46)(39,43,48,44)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=chi,d^a=dh,f^a=fh,f^b=fh,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253383);