Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499724 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,19)(17,29)(18,32)(20,24)(21,30)(22,28)(25,36)(26,42)(27,38)(31,40)(33,39)(34,41)(35,44)(37,43);(5,7,8,12)(6,9,10,11)(14,19)(15,24)(17,30)(18,28)(20,36)(21,38)(23,34)(25,41)(26,43)(27,44)(29,35)(31,33)(37,39)(40,42);(2,4)(13,15)(14,20)(16,25)(17,31)(18,33)(19,34)(21,39)(22,40)(23,29)(24,30)(26,28)(27,43)(32,37)(35,42)(36,38)(41,44);(6,10)(9,11)(14,19)(15,25)(21,35)(22,32)(24,41)(26,33)(29,38)(31,43)Polycyclic presentation:
2^9|b^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=di,f^c=fh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407173);