Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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×#2617 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,20)(17,18)(22,26)(23,27);(5,6,8,10)(7,9,12,11)(13,15,17,20)(14,16,18,19)(21,22,25,26)(23,24,27,28);(7,12)(9,11)(14,18)(16,19)(22,26)(23,27);(5,7,8,12)(6,9,10,11)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28);(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=fgh,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 | #72327351, #72327366, #72327425, … |
| maximal normal subgroup of | #72327351, #72327366, #72327425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 467902);