Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | |
2048 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)(27,39)(28,35)(29,36)(30,37)(31,38)(32,40)(41,43)(42,44);(5,7,8,12)(6,9,10,11)(13,15,19,28)(14,20,23,35)(16,29,31,41)(17,26,18,27)(21,36,38,43)(22,34,24,39)(25,32,42,30)(33,40,44,37);(1,2,3,4)(13,16,19,31)(14,21,23,38)(15,25,28,42)(17,30,18,32)(20,33,35,44)(22,37,24,40)(26,41,27,29)(34,43,39,36);(6,10)(9,11)(13,17,19,18)(14,22,23,24)(15,26,28,27)(16,30,31,32)(20,34,35,39)(21,37,38,40)(25,41,42,29)(33,43,44,36);(7,12)(9,11)(14,23)(15,27)(17,18)(20,34)(21,38)(25,29)(26,28)(30,32)(33,43)(35,39)(36,44)(41,42)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590100);