Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417597 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3395 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,13)(8,17)(9,12)(10,15)(11,18)(14,19)(16,20);(5,7,8,15,9,16,18,14)(6,10,11,13,12,19,17,20);(10,19)(13,20);(2,4)(7,14)(8,18)(10,20)(11,17)(13,19)(15,16)Polycyclic presentation:
2^9|b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214076, #72215972, #72215995, … |
| maximal normal subgroup of | #72214076, #72215972, #72215995, … |
| maximal quotient of | #72214129, #72214488, #72214883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324970);