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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2047 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)(13,14,20,36,19,24,18,25)(15,26,30,21,29,42,28,39)(16,22,37,44,31,40,32,38)(17,33,27,41,34,23,35,43);(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,21,24,39)(16,27,31,35)(17,32,34,37)(18,30,20,28)(22,23,40,33)(25,42,36,26)(38,43,44,41);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,31)(14,22,24,40)(15,27,29,35)(17,30,34,28)(18,32,20,37)(21,23,39,33)(25,38,36,44)(26,41,42,43);(7,12)(9,11)(13,17,20,27,19,34,18,35)(14,23,36,43,24,33,25,41)(15,16,30,37,29,31,28,32)(21,38,42,22,39,44,26,40);(6,10)(9,11)(14,24)(16,31)(17,34)(21,39)(25,36)(26,42)(27,35)(32,37)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,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, 6914385);