Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420407 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×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 | |
2200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(10,17)(13,19);(5,6)(7,13,14,19)(8,12)(9,11)(10,15,17,20)(16,18);(5,7)(6,10,11,17)(8,15)(9,14)(12,13,18,19)(16,20);(2,4)(8,16)(12,18)(13,19)(15,20)Polycyclic presentation:
2^9|b^2=e,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,e^c=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72239252, #72256724, #72256729, … |
| maximal normal subgroup of | #72239252, #72256724, #72256729, … |
| maximal quotient of | #72239267, C2×#422790 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329986);