Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2404 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3646 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)(5,6,10,12)(7,9,8,11)(14,18)(15,19)(21,22,24,26)(23,27,28,25);(5,7)(6,11)(8,10)(9,12)(13,14)(15,16)(17,18)(19,20)(21,22,24,26)(23,25,28,27);(1,2,4,3)(6,12)(7,8)(14,18)(15,19)(21,22)(23,25)(24,26)(27,28);(1,2,4,3)(5,8,10,7)(6,9,12,11)(13,15,17,19)(14,16,18,20)(21,22)(23,25)(24,26)(27,28);(1,3,4,2)(5,9)(6,7)(8,12)(10,11)(13,16)(14,19)(15,18)(17,20)(21,23,24,28)(22,25,26,27)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=gi,d^2=fgi,e^2=fghi,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72263773, #72263788, #72263797, … |
| maximal normal subgroup of | #72263773, #72263788, #72263797, … |
| maximal quotient of | #72293980, C2×#606390, #72328462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 513586);