Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2813 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,16,18)(15,19,20,17)(23,27)(26,28);(2,4)(5,7,9,8)(6,10,11,12)(13,15,16,20)(14,17,18,19)(21,22,25,24)(23,26,27,28);(1,2)(3,4)(7,8)(10,12)(14,18)(17,19)(21,23)(22,26)(24,28)(25,27);(5,8,9,7)(6,10,11,12)(13,14)(15,17)(16,18)(19,20)(21,24,25,22)(23,28,27,26);(5,6)(7,12)(8,10)(9,11)(13,15)(14,17)(16,20)(18,19)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72326988, #72326998, #72327002, … |
| maximal normal subgroup of | #72326988, #72326998, #72327002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1035951);