Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417597 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
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 | |
3047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,18)(16,20)(17,19);(1,2)(3,4)(5,7)(6,11)(8,17)(9,15)(10,16)(12,19)(13,18)(14,20);(6,12)(8,9)(11,18)(13,14)(15,17)(19,20);(2,4)(6,13)(8,9)(11,18)(12,14)(15,17)(19,20)Polycyclic presentation:
2^9|f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214076, #72216037, #72216047, … |
| maximal normal subgroup of | #72214076, #72216037, #72216047, … |
| maximal quotient of | C2×#417597, #72236737, #72239546, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324793);