Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #195990 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(9,10)(11,17)(12,15,20,22)(13,16)(14,19)(18,23,24,21);(1,2,4,3)(5,6)(7,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23);(6,8)(9,12)(10,15,16,22)(11,18)(13,20)(14,21,17,23)(19,24);(1,3,4,2)(10,16)(14,17)(15,22)(21,23)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,d^2=ei,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72313958, #72313964, #72314022, … |
| maximal normal subgroup of | #72313958, #72313964, #72314022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281542);