Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#535 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,11,9,7,10,12)(14,18)(16,19)(20,24)(21,25)(23,26)(27,28);(5,7,8,12,9,6,10,11)(13,14,19,25,17,21,16,18)(15,20,26,28,22,27,23,24);(1,2,3,4)(13,15)(14,20)(16,23)(17,22)(18,24)(19,26)(21,27)(25,28);(15,22)(20,27)(23,26)(24,28)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^c=dh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72215872, #72215902, #72215904 |
| maximal normal subgroup of | #72215872, #72215902, #72215904 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111238);