Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(5,6)(7,10)(8,11)(9,12);(2,4)(5,7,9,8)(6,10,12,11)(14,18)(15,16)(19,20);(5,7,9,8)(6,11,12,10)(13,14)(15,18)(16,20)(17,19);(5,8,9,7)(6,11,12,10)(13,14)(15,18)(16,20)(17,19);(13,15,17,16)(14,18,19,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72305415, #72305417, #72305419 |
| maximal normal subgroup of | #72305415, #72305417, #72305419 |
| maximal quotient of | #72306170 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360282);