Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682420 |
| 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 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,11)(8,10)(9,12)(13,14,18,20)(15,16,19,17);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11);(2,4)(13,15,17,14,18,19,16,20);(6,10)(9,11)(13,16,18,17)(14,19,20,15);(2,4)(7,12)(9,11)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282754, #72282796, #72282887 |
| maximal normal subgroup of | #72282754, #72282796, #72282887 |
| maximal quotient of | #72319895, #72319983 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589623);