Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35687 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2655 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14,18,23)(15,25,28,20)(16,30,27,43)(17,33,26,44)(19,37,36,24)(21,34,40,32)(22,31,41,35)(29,39,38,42);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,31)(17,34)(18,28)(19,38)(21,33)(22,30)(23,25)(24,42)(26,32)(27,35)(29,36)(37,39)(40,44)(41,43);(13,16,18,27)(14,21,23,40)(15,26,28,17)(19,34,36,32)(20,22,25,41)(24,43,37,30)(29,35,38,31)(33,39,44,42);(2,4)(13,17,18,26)(14,22,23,41)(15,27,28,16)(19,31,36,35)(20,21,25,40)(24,44,37,33)(29,32,38,34)(30,39,43,42);(6,10)(9,11)(14,23)(15,28)(16,27)(22,41)(24,37)(29,38)(32,34)(33,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421782);