Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14,19,24)(15,26,30,39)(16,32,17,22)(18,25,20,23)(21,31,38,29)(27,43,28,40)(33,44,36,37)(34,41,35,42);(1,2,3,4)(13,15,18,29,19,30,20,31)(14,21,23,26,24,38,25,39)(16,33,35,27,17,36,34,28)(22,40,42,37,32,43,41,44);(6,9)(10,12)(13,16,19,17)(14,22,24,32)(15,27,30,28)(18,34,20,35)(21,37,38,44)(23,41,25,42)(26,40,39,43)(29,33,31,36);(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,22,24,32)(15,28,30,27)(18,35,20,34)(21,37,38,44)(23,41,25,42)(26,40,39,43)(29,36,31,33);(13,18,19,20)(14,23,24,25)(15,29,30,31)(16,34,17,35)(21,26,38,39)(22,41,32,42)(27,33,28,36)(37,40,44,43)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^c=fi,h^c=hi,f^d=fi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357329);