Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1817 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)(13,14,19,23,20,24,18,17)(15,25,28,39,29,38,27,21)(16,30,33,35,34,22,36,32)(26,42,43,37,31,41,44,40);(5,6)(7,11)(8,10)(9,12)(13,15,20,29)(14,21,24,39)(16,31,34,26)(17,25,23,38)(18,28,19,27)(22,40,30,37)(32,41,35,42)(33,43,36,44);(2,4)(5,7,8,12)(6,9,10,11)(13,16,20,34)(14,22,24,30)(15,26,29,31)(17,35,23,32)(18,36,19,33)(21,37,39,40)(25,41,38,42)(27,44,28,43);(7,12)(9,11)(13,17,18,24,20,23,19,14)(15,21,27,38,29,39,28,25)(16,32,36,22,34,35,33,30)(26,40,44,41,31,37,43,42);(6,10)(9,11)(13,18,20,19)(14,17,24,23)(15,27,29,28)(16,33,34,36)(21,38,39,25)(22,32,30,35)(26,43,31,44)(37,41,40,42)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=h,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972102);