Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
758 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)(6,9)(10,11)(13,14)(15,20)(16,26)(17,30)(18,34)(19,23)(21,28)(22,32)(24,40)(25,39)(27,38)(29,43)(31,35)(33,42)(36,44)(37,41);(5,6,8,9)(7,11,12,10)(13,15,19,18)(14,20,23,34)(16,27,29,24)(17,25,33,32)(21,36,37,35)(22,30,39,42)(26,38,43,40)(28,44,41,31);(13,16,19,29)(14,21,23,37)(15,24,18,27)(17,31,33,44)(20,35,34,36)(22,38,39,40)(25,41,32,28)(26,42,43,30);(5,7,8,12)(6,10,9,11)Polycyclic presentation:
2^9|a^2=g,b^2=ei,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fh,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404811);