Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2008 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)(7,11)(10,12)(13,14,19,25)(15,20,29,38)(16,30,32,22)(17,24,18,23)(21,34,40,31)(26,36,27,44)(28,41,33,37)(35,42,39,43);(1,2)(3,4)(6,9)(7,11)(13,15,19,29)(14,20,25,38)(16,27,32,26)(17,33,18,28)(21,39,40,35)(22,36,30,44)(23,41,24,37)(31,42,34,43);(5,6)(7,10)(8,9)(11,12)(13,16,19,32)(14,21,25,40)(15,26,29,27)(17,31,18,34)(20,35,38,39)(22,23,30,24)(28,42,33,43)(36,37,44,41);(2,4)(6,9)(10,12)(13,16)(14,22)(15,27)(17,34)(18,31)(19,32)(20,36)(21,24)(23,40)(25,30)(26,29)(28,42)(33,43)(35,41)(37,39)(38,44);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,25,24)(15,28,29,33)(16,31,32,34)(20,37,38,41)(21,30,40,22)(26,42,27,43)(35,44,39,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050147);