Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 | #39131 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1860 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)(5,6,10,9)(7,12,8,11)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,28)(21,33)(26,34)(27,36)(29,35)(31,38)(32,37)(39,41)(40,44)(42,43);(5,7,10,8)(6,11,9,12)(13,15,17,28,19,29,18,27)(14,20,23,25,24,36,22,35)(16,26,32,42,31,40,33,41)(21,34,38,39,37,44,30,43);(2,4)(5,6,10,9)(7,12,8,11)(13,16,17,32,19,31,18,33)(14,21,23,38,24,37,22,30)(15,26,28,42,29,40,27,41)(20,34,25,39,36,44,35,43);(5,8,10,7)(6,11,9,12)(16,31)(21,37)(26,40)(30,38)(32,33)(34,44)(39,43)(41,42);(5,9,10,6)(7,12,8,11)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,32,31,33)(20,25,36,35)(21,30,37,38)(26,41,40,42)(34,39,44,43)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6474798);