Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431710 |
| 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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,10)(7,11,12,9)(13,14,17,22,18,23,30,19)(15,24,26,34,27,37,31,20)(16,21,29,36,28,35,41,32)(25,38,40,43,39,44,42,33);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,19)(17,30)(20,24)(21,32)(22,23)(26,31)(29,41)(33,38)(34,37)(35,36)(40,42)(43,44);(2,4)(13,15,18,27)(14,20,23,34)(16,25,28,39)(17,31,30,26)(19,24,22,37)(21,33,35,43)(29,42,41,40)(32,38,36,44);(13,16)(14,21)(15,25)(17,29)(18,28)(19,32)(20,33)(22,36)(23,35)(24,38)(26,40)(27,39)(30,41)(31,42)(34,43)(37,44);(16,28)(21,35)(25,39)(29,41)(32,36)(33,43)(38,44)(40,42)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339622);