Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19);(2,4)(5,7,8,6)(12,13)(16,21)(18,23)(19,22);(1,2,3,4)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(9,12)(10,16)(11,19)(13,14)(15,23)(17,21)(18,24)(20,22)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^b=fi,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261364, #72261411, #72261896, … |
| maximal normal subgroup of | #72261364, #72261411, #72261896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275294);