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 | #35757 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2679 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)(13,14,19,24)(15,20,25,38)(16,27,30,43)(17,31,32,44)(18,33,37,23)(21,35,40,29)(22,36,41,28)(26,42,34,39);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,28)(17,29)(18,34)(19,25)(21,31)(22,27)(23,42)(24,38)(26,37)(30,36)(32,35)(33,39)(40,44)(41,43);(13,16,19,30)(14,21,24,40)(15,17,25,32)(18,35,37,29)(20,22,38,41)(23,43,33,27)(26,36,34,28)(31,39,44,42);(2,4)(13,17)(14,22)(15,16)(18,36)(19,32)(20,21)(23,44)(24,41)(25,30)(26,35)(27,42)(28,37)(29,34)(31,33)(38,40)(39,43);(6,10)(9,11)(15,25)(17,32)(20,38)(22,41)(26,34)(28,36)(31,44)(39,42)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^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,d^c=di,g^c=gi,h^c=hi,e^d=ei,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, 7421996);