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 | |
2039 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)(13,14,20,33,19,25,18,24)(15,21,34,43,29,38,28,37)(16,23,30,22,17,32,31,39)(26,36,40,35,27,42,41,44);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,21,25,38)(16,26,17,27)(18,28,20,34)(22,35,39,44)(23,36,32,42)(24,37,33,43)(30,40,31,41);(5,7)(6,9)(8,12)(10,11)(13,16,19,17)(14,22,25,39)(15,26,29,27)(18,30,20,31)(21,35,38,44)(23,24,32,33)(28,40,34,41)(36,37,42,43);(1,3,4,2)(7,12)(9,11)(13,17,19,16)(14,23,25,32)(15,27,29,26)(18,30,20,31)(21,36,38,42)(22,24,39,33)(28,40,34,41)(35,37,44,43);(6,10)(9,11)(13,18,19,20)(14,24,25,33)(15,28,29,34)(16,30,17,31)(21,37,38,43)(22,32,39,23)(26,40,27,41)(35,42,44,36)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=fi,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, 6913682);