Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| 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 | |
1220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,11,15,10,16)(7,12,24,28,18,29,19,27)(8,20,22,13,23,31,25,32)(17,33,35,30,21,36,34,26);(1,2)(3,4)(5,7,9,19,11,18,10,24)(6,12,16,28,15,29,14,27)(8,21,22,35,23,17,25,34)(13,30,20,36,32,26,31,33);(2,4)(5,8)(6,13,15,32)(7,17,18,21)(9,22)(10,25)(11,23)(12,26)(14,31,16,20)(19,34,24,35)(27,36)(28,33)(29,30);(6,14)(7,18)(9,10)(12,27)(13,31)(15,16)(17,21)(20,32)(22,25)(26,36)(28,29)(30,33)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329099);