Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 | #36078 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,19,18)(15,20,26,25)(16,21,29,33)(17,22,32,31)(23,34,38,27)(24,35,41,40)(28,36,43,30)(37,39,44,42);(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,27)(17,24)(18,25)(19,26)(21,23)(22,35)(28,42)(29,34)(30,44)(31,40)(32,41)(33,38)(36,37)(39,43);(2,4)(13,16,19,29)(14,21,18,33)(15,23,26,38)(17,30,32,36)(20,34,25,27)(22,28,31,43)(24,39,41,42)(35,44,40,37);(5,8,9,7)(6,10,11,12)(13,17,19,32)(14,22,18,31)(15,24,26,41)(16,28,29,43)(20,35,25,40)(21,36,33,30)(23,37,38,44)(27,42,34,39);(14,18)(15,25)(17,31)(20,26)(21,33)(22,32)(23,27)(28,36)(30,43)(34,38)(35,40)(37,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483702);