Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,15,10,16,9,24)(7,17,20,12,21,32,25,29)(8,14,22,30,23,13,26,31)(18,33,36,28,19,35,34,27);(1,2)(3,4)(5,7,11,20,10,21,9,25)(6,12,15,32,16,29,24,17)(8,19,22,34,23,18,26,36)(13,27,31,33,14,28,30,35);(2,4)(5,8,9,22,10,23,11,26)(6,13,15,30,16,14,24,31)(7,18,20,34,21,19,25,36)(12,27,17,33,29,28,32,35);(5,8)(6,14)(7,19)(9,26)(10,23)(11,22)(12,28)(13,16)(15,30)(17,33)(18,21)(20,34)(24,31)(25,36)(27,29)(32,35)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364100);