Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
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)(16,29,31,38)(17,23,18,22)(21,32,37,30)(25,39,41,44)(26,36,28,34)(33,42,43,40);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,25)(17,26)(18,28)(19,27)(21,33)(22,34)(23,36)(24,35)(29,39)(30,40)(31,41)(32,42)(37,43)(38,44);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,30)(18,32)(19,31)(20,33)(22,29)(23,38)(24,37)(26,40)(27,41)(28,42)(34,39)(35,43)(36,44);(2,4)(13,17,19,18)(14,22,24,23)(15,26,27,28)(16,30,31,32)(20,34,35,36)(21,29,37,38)(25,40,41,42)(33,39,43,44);(15,27)(20,35)(25,41)(26,28)(33,43)(34,36)(39,44)(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=dh,e^b=ei,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345515);