Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2580 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
973 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)(5,6,11,16)(7,18)(8,23,26,13)(9,14,29,27)(10,15,25,17)(12,30)(19,34)(20,35)(21,33)(22,32)(24,28)(31,36);(5,7)(6,12)(8,24)(9,28)(10,30)(11,22)(13,34)(14,35)(15,32)(16,33)(17,18)(19,27)(20,23)(21,25)(26,31)(29,36);(5,8,25,27,11,26,10,14)(6,13,17,29,16,23,15,9)(7,19,21,31,22,35,30,24)(12,28,18,34,33,36,32,20);(2,4)(5,9,10,23,11,29,25,13)(6,14,15,26,16,27,17,8)(7,20,21,28,22,34,30,36)(12,31,18,35,33,24,32,19)Polycyclic presentation:
2^9|a^2=f,c^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 | #72322946 |
| maximal normal subgroup of | #72322946 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332593);