Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417579 |
| 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 | |
3034 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)(6,11)(8,9)(12,15)(13,20)(14,19)(16,18);(5,6,8,11,10,14,9,19)(7,15,16,12,17,20,18,13);(5,7)(6,12)(8,16)(9,18)(10,17)(11,20)(13,14)(15,19);(2,4)(5,7)(6,13)(8,18)(9,16)(10,17)(11,20)(12,14)(15,19)Polycyclic presentation:
2^9|b^2=e,e^2=i,g^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,g^b=gi,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72213224, #72213250, #72213286, … |
| maximal normal subgroup of | #72213224, #72213250, #72213286, … |
| maximal quotient of | #72226268, C2×#417798, #72258125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324994);