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 |
| prime radical | #45265 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,24,19,25,17,23)(15,20,28,37,29,38,27,36)(16,30,31,39,32,22,33,21)(26,40,41,44,42,35,43,34);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,20,25,38)(16,26,32,42)(17,27,18,28)(21,34,39,44)(22,35,30,40)(23,36,24,37)(31,41,33,43);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,26)(17,31)(18,33)(19,32)(20,34)(22,24)(23,30)(25,39)(27,41)(28,43)(29,42)(35,37)(36,40)(38,44);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,26)(17,33)(18,31)(19,32)(20,35)(21,24)(23,39)(25,30)(27,43)(28,41)(29,42)(34,37)(36,44)(38,40);(6,10)(9,11)(13,17,19,18)(14,23,25,24)(15,27,29,28)(16,31,32,33)(20,36,38,37)(21,30,39,22)(26,41,42,43)(34,40,44,35)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 6971668);