Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,8,9)(7,11,12,10)(13,14,17,20)(15,16,19,18)(21,22,25,24)(23,27,28,26);(1,3,4,2)(6,9)(7,12)(13,15,17,19)(14,18,20,16)(22,24)(26,27);(5,7,8,12)(6,10,9,11)(13,16,17,18)(14,19,20,15)(21,22,25,24)(23,27,28,26);(5,7)(6,10)(8,12)(9,11)(13,16)(14,15)(17,18)(19,20)(21,23,25,28)(22,26,24,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,19)(18,20)(21,24,25,22)(23,27,28,26)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=f,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72326687, #72326706, #72326730, … |
| maximal normal subgroup of | #72326687, #72326706, #72326730, … |
| maximal quotient of | #72329010, #72332727 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 469675);