Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470161 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4035 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)(7,10)(8,11)(9,12)(13,14)(15,25)(16,30)(17,24)(18,23)(19,22)(20,27)(21,32)(26,38)(28,36)(29,37)(31,35)(33,39)(34,40)(41,42)(43,44);(2,4)(7,8)(10,11)(13,15,19,29)(14,20,22,36)(16,31,34,26)(17,28,18,27)(21,38,39,35)(23,37,24,25)(30,41,40,44)(32,42,33,43);(5,7,9,8)(6,10,12,11)(13,16,19,34)(14,21,22,39)(15,26,29,31)(17,32,18,33)(20,35,36,38)(23,30,24,40)(25,41,37,44)(27,42,28,43);(5,8,9,7)(6,11,12,10)(14,22)(20,36)(21,39)(23,24)(25,37)(30,40)(35,38)(41,44);(6,12)(10,11)(13,17,19,18)(14,23,22,24)(15,27,29,28)(16,32,34,33)(20,25,36,37)(21,30,39,40)(26,42,31,43)(35,41,38,44)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394819);