Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,16)(7,17,21,28)(8,22,25,32)(9,15,10,14)(12,19,29,20)(13,26,31,24)(18,30,23,27)(33,34,36,35);(1,2)(3,4)(5,7,10,19,11,21,9,20)(6,12,15,28,16,29,14,17)(8,23,24,34,25,18,26,35)(13,30,22,36,31,27,32,33);(5,8)(6,13)(7,18)(9,24)(10,26)(11,25)(12,27)(14,22)(15,32)(16,31)(17,33)(19,34)(20,35)(21,23)(28,36)(29,30);(2,4)(7,19)(9,10)(12,28)(14,15)(17,29)(18,34)(20,21)(22,32)(23,35)(24,26)(27,36)(30,33)Polycyclic presentation:
2^9|a^2=i,b^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327970);