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×#2017 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16,12,17,10,27)(7,18,21,13,22,30,28,31)(8,23,20,32,25,36,24,14)(9,15,26,34,19,35,29,33);(1,2)(3,4)(5,7,11,21,12,22,10,28)(6,13,16,30,17,31,27,18)(8,9,20,26,25,19,24,29)(14,15,23,34,32,35,36,33);(2,4)(5,8)(6,14)(7,19)(9,22)(10,20)(11,24)(12,25)(13,15)(16,36)(17,32)(18,34)(21,26)(23,27)(28,29)(30,33)(31,35);(5,9,12,19)(6,15,17,35)(7,20,22,24)(8,21,25,28)(10,29,11,26)(13,23,31,36)(14,30,32,18)(16,34,27,33)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363583);