Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(14,19)(15,23)(16,27)(17,30)(20,37)(21,41)(22,36)(24,40)(25,38)(26,33)(28,32)(31,34)(35,39)(42,43);(5,7,9,8)(6,10,11,12)(13,14,18,22)(15,24,26,37)(16,28,31,41)(17,32,30,21)(19,34,36,27)(20,38,40,25)(23,43,33,35)(29,42,44,39);(5,8,9,7)(6,10,11,12);(13,15,18,26)(14,20,22,40)(16,29,31,44)(17,33,30,23)(19,35,36,43)(21,42,32,39)(24,41,37,28)(25,34,38,27)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405190);