Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2482 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,16,20,27)(8,13)(9,14)(10,24)(11,15)(12,19,28,18)(17,29,35,33)(21,26,34,36)(22,30)(23,32)(25,31);(5,7)(6,12)(8,21)(9,18)(10,19)(11,20)(13,29)(14,27)(15,28)(16,24)(17,22)(23,34)(25,35)(26,31)(30,36)(32,33);(5,8,9,22,11,23,10,25)(6,13,14,30,15,32,24,31)(7,17,18,34,20,35,19,21)(12,26,27,29,28,36,16,33);(2,4)(7,18,20,19)(12,27,28,16)(17,34,35,21)(26,29,36,33)Polycyclic presentation:
2^9|a^2=f,c^2=ef,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321897 |
| maximal normal subgroup of | #72321897 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 338368);