Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471338 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16788 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12,6,13,7,24)(3,14,18,26,17,9,19,30)(4,10,23,28,5,22,21,11)(15,25,20,32,31,29,16,27);(1,3,7,18,6,17,8,19)(2,9,12,26,13,14,24,30)(4,20,23,15,5,16,21,31)(10,27,11,25,22,32,28,29);(1,4)(2,10)(3,15,17,31)(5,6)(7,21)(8,23)(9,25,14,29)(11,24)(12,28)(13,22)(16,18,20,19)(26,27,30,32);(1,5,6,4)(2,11)(3,16,17,20)(7,21,8,23)(9,25)(10,24)(12,22)(13,28)(14,29)(15,18,31,19)(26,32)(27,30)Polycyclic presentation:
2^9|a^2=eg,b^2=efh,c^2=f,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=de,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237063, #72237066, #72237086, … |
| maximal normal subgroup of | #72237063, #72237066, #72237086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378534);