Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419041 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| 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 | |
1923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,17,19,12)(8,22,24,32)(9,15,10,14)(13,26,31,25)(18,30,34,36)(20,28,21,29)(23,27,35,33);(1,2)(3,4)(5,7,9,20,11,19,10,21)(6,12,14,28,16,17,15,29)(8,23,25,18,24,35,26,34)(13,30,22,27,31,36,32,33);(2,4)(5,8)(6,13)(7,18)(9,26)(10,25)(11,24)(12,27)(14,32)(15,22)(16,31)(17,33)(19,34)(20,23)(21,35)(28,30)(29,36);(7,19)(9,10)(12,17)(14,15)(18,34)(22,32)(25,26)(27,33)Polycyclic presentation:
2^9|a^2=i,b^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328054);