Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1805 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,17,19,23,18,24)(15,25,35,21,27,40,28,38)(16,29,36,31,32,22,33,34)(26,42,44,39,30,41,43,37);(5,6)(7,11)(8,10)(9,12)(13,15,19,27)(14,21,23,38)(16,30,32,26)(17,25,24,40)(18,35,20,28)(22,39,29,37)(31,41,34,42)(33,43,36,44);(2,4)(5,7,8,12)(6,9,10,11)(13,16,19,32)(14,22,23,29)(15,26,27,30)(17,34,24,31)(18,33,20,36)(21,37,38,39)(25,41,40,42)(28,43,35,44);(7,12)(9,11)(13,17,20,23,19,24,18,14)(15,21,35,40,27,38,28,25)(16,31,36,22,32,34,33,29)(26,39,44,41,30,37,43,42);(6,10)(9,11)(14,23)(15,27)(16,32)(17,24)(28,35)(33,36)(37,39)(41,42)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972245);