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 | #45093 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,17,19,24,18,23)(15,25,27,21,29,38,28,36)(16,30,32,37,34,22,33,31)(26,40,43,44,42,39,41,35);(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,21,24,36)(16,26,34,42)(17,25,23,38)(18,27,20,28)(22,35,30,44)(31,39,37,40)(32,41,33,43);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,26)(17,31)(18,33)(19,34)(20,32)(21,35)(23,37)(24,30)(25,39)(27,43)(28,41)(29,42)(36,44)(38,40);(7,12)(9,11)(13,17,20,24,19,23,18,14)(15,21,27,38,29,36,28,25)(16,31,32,30,34,37,33,22)(26,35,43,40,42,44,41,39);(6,10)(9,11)(13,18,19,20)(14,17,24,23)(15,27,29,28)(16,32,34,33)(21,25,36,38)(22,37,30,31)(26,41,42,43)(35,40,44,39)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=fi,e^2=i,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, 6914366);