Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 | #36009 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,21)(17,22)(18,24)(19,23)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40)(41,44)(42,43);(5,7)(6,9)(8,12)(10,11)(13,15,19,28)(14,20,23,36)(16,29,31,42)(17,27,18,26)(21,37,39,43)(22,35,24,34)(25,32,41,30)(33,40,44,38);(1,2,3,4)(13,16,19,31)(14,21,23,39)(15,25,28,41)(17,30,18,32)(20,33,36,44)(22,38,24,40)(26,29,27,42)(34,37,35,43);(6,10)(9,11)(13,17,19,18)(14,22,23,24)(15,26,28,27)(16,30,31,32)(20,34,36,35)(21,38,39,40)(25,29,41,42)(33,37,44,43);(7,12)(9,11)(14,23)(15,26)(17,18)(20,35)(21,39)(25,29)(27,28)(30,32)(33,43)(34,36)(37,44)(41,42)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589524);