Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2005 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)(7,11)(8,10)(9,12)(13,14,18,22)(15,19,24,32)(16,26,27,41)(17,29,30,21)(20,34,35,28)(23,36,37,44)(25,40,39,33)(31,42,43,38);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,23)(17,25)(18,24)(20,31)(21,33)(22,32)(26,36)(27,37)(28,38)(29,40)(30,39)(34,42)(35,43)(41,44);(2,4)(13,16,17,28,18,27,30,34)(14,20,21,26,22,35,29,41)(15,23,25,38,24,37,39,42)(19,31,33,36,32,43,40,44);(6,10)(9,11)(15,24)(16,27)(19,32)(20,35)(25,39)(26,41)(28,34)(33,40);(2,4)(7,12)(9,11)(16,27)(20,35)(23,37)(26,41)(28,34)(31,43)(36,44)(38,42)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590409);