Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,13,16,10)(8,17)(9,11)(12,19)(14,18,20,15);(2,4)(5,6)(7,14)(8,17)(9,11)(10,15)(12,19)(13,18)(16,20);(5,7)(6,10)(8,18)(9,16)(11,13)(12,20)(14,17)(15,19);(2,4)Polycyclic presentation:
2^9|a^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,h^c=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72219483, #72233855, #72234086, … |
| maximal normal subgroup of | #72219483, #72233855, #72234086, … |
| maximal quotient of | #72213681, #72213874, #72214520 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266675);