Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2333 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)(13,14,17,21)(15,23,24,19)(16,20,25,27)(18,22,26,28);(5,7,9,8)(6,10,12,11)(13,15)(14,19)(16,18)(17,24)(20,28)(21,23)(22,27)(25,26);(2,4)(5,7,9,8)(6,11,12,10)(15,16)(18,26)(19,27)(20,23)(22,28)(24,25);(5,8,9,7)(6,10,12,11)(13,16,17,25)(14,20,21,27)(15,18,24,26)(19,28,23,22);(14,21)(15,16)(18,26)(19,20)(23,27)(24,25)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fi,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=ehi,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307116, #72307157, #72307466, … |
| maximal normal subgroup of | #72307116, #72307157, #72307466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484658);