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 | #2487 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
866 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)(5,6,10,15,11,16,12,29)(7,17,20,32,22,34,21,13)(8,23,25,28,26,14,30,9)(18,19,24,27,35,33,36,31);(5,7,12,20,11,22,10,21)(6,13,29,17,16,32,15,34)(8,24,30,35,26,36,25,18)(9,27,14,33,28,31,23,19);(5,8)(6,14)(7,18)(9,15)(10,25)(11,26)(12,30)(13,31)(16,23)(17,19)(20,24)(21,36)(22,35)(27,32)(28,29)(33,34);(2,4)(5,9,10,23,11,28,12,14)(6,8,15,25,16,26,29,30)(7,19,20,27,22,33,21,31)(13,18,17,24,32,35,34,36)Polycyclic presentation:
2^9|a^2=e,b^2=ef,d^2=e,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 | #72322778 |
| maximal normal subgroup of | #72322778 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328684);