Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C25×C42 |
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 | |
4430 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)(6,9)(10,11)(14,17)(15,19)(22,24)(23,27);(5,6,8,9)(7,11,12,10)(13,14,16,17)(15,18,19,20)(21,22,25,24)(23,28,27,26);(13,15)(14,18)(16,19)(17,20)(21,23)(22,26)(24,28)(25,27);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,24,25,22)(23,28,27,26);(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=hi,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72293921, C2×#557860, #72323465, … |
| maximal normal subgroup of | #72293921, C2×#557860, #72323465, … |
| maximal quotient of | #72293980, C2×#557860, #72323663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 465056);