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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2217 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)(5,6)(7,9)(8,10)(11,12)(13,14,18,24,19,23,34,37)(15,25,28,22,27,41,17,20)(16,29,33,44,32,21,38,40)(26,43,31,36,30,42,35,39);(5,6)(7,11)(8,10)(9,12)(13,15,19,27)(14,20,23,22)(16,30,32,26)(17,34,28,18)(21,39,29,36)(24,41,37,25)(31,38,35,33)(40,42,44,43);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,32)(14,21,23,29)(15,26,27,30)(17,35,28,31)(18,33,34,38)(20,36,22,39)(24,40,37,44)(25,42,41,43);(7,12)(9,11)(13,17,18,15,19,28,34,27)(14,22,24,41,23,20,37,25)(16,31,33,30,32,35,38,26)(21,36,40,42,29,39,44,43);(6,10)(9,11)(14,23)(15,27)(16,32)(17,28)(24,37)(33,38)(36,39)(42,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=f,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, 6928947);