Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #42796 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1883 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)(14,20)(17,18)(21,35)(22,36)(23,24)(25,40)(27,34)(29,39)(31,32)(37,41)(38,44)(42,43);(5,6)(7,11)(8,10)(9,12)(13,14,18,20,19,24,17,23)(15,25,27,40,28,21,34,35)(16,29,32,39,33,36,31,22)(26,41,42,37,30,44,43,38);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,30)(17,34)(18,27)(19,28)(20,35)(22,38)(23,40)(24,25)(26,33)(29,41)(31,42)(32,43)(36,44)(37,39);(7,12)(9,11)(13,16,19,33)(14,22,24,39)(15,26,28,30)(17,32,18,31)(20,36,23,29)(21,37,25,38)(27,43,34,42)(35,41,40,44);(6,10)(9,11)(13,17,19,18)(14,23,24,20)(15,27,28,34)(16,31,33,32)(21,35,25,40)(22,36,39,29)(26,42,30,43)(37,44,38,41)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910518);