Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2076 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)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,27,35)(16,28,30,38)(17,23,18,22)(21,32,36,29)(25,39,41,43)(26,37,31,34)(33,40,44,42);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,25)(17,31)(18,26)(19,27)(21,33)(22,37)(23,34)(24,35)(28,39)(29,42)(30,41)(32,40)(36,44)(38,43);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,29)(18,32)(19,30)(20,33)(22,28)(23,38)(24,36)(26,40)(27,41)(31,42)(34,43)(35,44)(37,39);(1,2)(3,4)(13,17,19,18)(14,22,24,23)(15,26,27,31)(16,29,30,32)(20,34,35,37)(21,28,36,38)(25,40,41,42)(33,43,44,39);(2,4)(15,27)(20,35)(25,41)(26,31)(33,44)(34,37)(39,43)(40,42)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482144);