Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(9,10,12,11)(14,16)(15,20)(17,18);(5,6)(7,8)(10,11)(13,14)(15,17)(16,18)(19,20);(2,4)(6,8)(9,10,12,11)(13,15,17,14,19,16,18,20);(9,11,12,10)(13,16,18,14,19,15,17,20)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72305522, #72305558, #72305590, … |
| maximal normal subgroup of | #72305522, #72305558, #72305590, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327706);