Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,18,22,19,23,17,30)(15,20,25,36,26,35,31,34)(16,27,28,38,29,37,32,21)(24,39,42,43,41,44,40,33);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,24)(17,31)(18,25)(19,26)(21,33)(22,36)(23,35)(27,39)(28,42)(29,41)(30,34)(32,40)(37,44)(38,43);(6,9)(10,12)(13,16,19,29)(14,21,23,38)(15,24,26,41)(17,28,18,32)(20,33,35,43)(22,37,30,27)(25,40,31,42)(34,39,36,44);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,30)(15,25,26,31)(16,28,29,32)(20,34,35,36)(21,37,38,27)(24,40,41,42)(33,39,43,44);(7,11)(10,12)(15,26)(20,35)(24,41)(25,31)(33,43)(34,36)(39,44)(40,42)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ei,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6452303);